Micron Document
<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Variance function</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Variance_function"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Variance_function rootpage-Variance_function skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Variance function</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">
<style data-mw-deduplicate="TemplateStyles:r1305433154">
/* start https://en.wikipedia.org/ */


.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}


/* end https://en.wikipedia.org/ */
</style>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
/* start https://en.wikipedia.org/ */


.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}


/* end https://en.wikipedia.org/ */
</style><div role="note" class="hatnote navigation-not-searchable">For variance as a function of space-time separation, see <a href="Variogram" title="Variogram">Variogram</a>.</div>
<style data-mw-deduplicate="TemplateStyles:r1129693374">
/* start https://en.wikipedia.org/ */


.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}


/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1246091330">
/* start https://en.wikipedia.org/ */


.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}


/* end https://en.wikipedia.org/ */
</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th></tr><tr><th class="sidebar-heading">
Models</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>
<li><a href="Simple_linear_regression" title="Simple linear regression">Simple regression</a></li>
<li><a href="Polynomial_regression" title="Polynomial regression">Polynomial regression</a></li>
<li><a href="General_linear_model" title="General linear model">General linear model</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></li>
<li><a href="Vector_generalized_linear_model" title="Vector generalized linear model">Vector generalized linear model</a></li>
<li><a href="Discrete_choice" title="Discrete choice">Discrete choice</a></li>
<li><a href="Binomial_regression" title="Binomial regression">Binomial regression</a></li>
<li><a href="Binary_regression" title="Binary regression">Binary regression</a></li>
<li><a href="Logistic_regression" title="Logistic regression">Logistic regression</a></li>
<li><a href="Multinomial_logistic_regression" title="Multinomial logistic regression">Multinomial logistic regression</a></li>
<li><a href="Mixed_logit" title="Mixed logit">Mixed logit</a></li>
<li><a href="Probit_model" title="Probit model">Probit</a></li>
<li><a href="Multinomial_probit" title="Multinomial probit">Multinomial probit</a></li>
<li><a href="Ordered_logit" title="Ordered logit">Ordered logit</a></li>
<li><a href="Ordered_probit" class="mw-redirect" title="Ordered probit">Ordered probit</a></li>
<li><a href="Poisson_regression" title="Poisson regression">Poisson</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Multilevel_model" title="Multilevel model">Multilevel model</a></li>
<li><a href="Fixed_effects_model" title="Fixed effects model">Fixed effects</a></li>
<li><a href="Random_effects_model" title="Random effects model">Random effects</a></li>
<li><a href="Mixed_model" title="Mixed model">Linear mixed-effects model</a></li>
<li><a href="Nonlinear_mixed-effects_model" title="Nonlinear mixed-effects model">Nonlinear mixed-effects model</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>
<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>
<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>
<li><a href="Robust_regression" title="Robust regression">Robust</a></li>
<li><a href="Quantile_regression" title="Quantile regression">Quantile</a></li>
<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>
<li><a href="Principal_component_regression" title="Principal component regression">Principal components</a></li>
<li><a href="Least-angle_regression" title="Least-angle regression">Least angle</a></li>
<li><a href="Local_regression" title="Local regression">Local</a></li>
<li><a href="Segmented_regression" title="Segmented regression">Segmented</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">Errors-in-variables</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Estimation</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Least_squares" title="Least squares">Least squares</a></li>
<li><a href="Linear_least_squares" title="Linear least squares">Linear</a></li>
<li><a href="Non-linear_least_squares" title="Non-linear least squares">Non-linear</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary</a></li>
<li><a href="Weighted_least_squares" title="Weighted least squares">Weighted</a></li>
<li><a href="Generalized_least_squares" title="Generalized least squares">Generalized</a></li>
<li><a href="Generalized_estimating_equation" title="Generalized estimating equation">Generalized estimating equation</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Partial_least_squares_regression" title="Partial least squares regression">Partial</a></li>
<li><a href="Total_least_squares" title="Total least squares">Total</a></li>
<li><a href="Non-negative_least_squares" title="Non-negative least squares">Non-negative</a></li>
<li><a href="Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Ridge regression</a></li>
<li><a href="Regularized_least_squares" title="Regularized least squares">Regularized</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li>
<li><a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">Iteratively reweighted</a></li>
<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li>
<li><a href="Bayesian_multivariate_linear_regression" title="Bayesian multivariate linear regression">Bayesian multivariate</a></li>
<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Background</th></tr><tr><td class="sidebar-content">
<ul><li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>
<li><a href="Mean_and_predicted_response" class="mw-redirect" title="Mean and predicted response">Mean and predicted response</a></li>
<li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>
<li><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></li>
<li><a href="Studentized_residual" title="Studentized residual">Studentized residual</a></li>
<li><a href="Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a></li></ul></td>
</tr><tr><td class="sidebar-below">
<ul><li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
/* start https://en.wikipedia.org/ */


.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}


/* end https://en.wikipedia.org/ */
</style></td></tr></tbody></table>
<p>In <a href="Statistics" title="Statistics">statistics</a>, the <b>variance function</b> is a <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a> that depicts the <a href="Variance" title="Variance">variance</a> of a <a href="Random_quantity" class="mw-redirect" title="Random quantity">random quantity</a> as a function of its <a href="Mean" title="Mean">mean</a>. The variance function is a measure of <a href="Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">heteroscedasticity</a> and plays a large role in many settings of statistical modelling. It is a main ingredient in the <a href="Generalized_linear_model" title="Generalized linear model">generalized linear model</a> framework and a tool used in <a href="Non-parametric_regression" class="mw-redirect" title="Non-parametric regression">non-parametric regression</a>,<sup id="cite_ref-Muller1_1-0" class="reference"><a href="#cite_note-Muller1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Semiparametric_regression" title="Semiparametric regression">semiparametric regression</a><sup id="cite_ref-Muller1_1-1" class="reference"><a href="#cite_note-Muller1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <a href="Functional_data_analysis" title="Functional data analysis">functional data analysis</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity. In a non-parametric setting, the variance function is assumed to be a <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Intuition">Intuition</h2></div>
<p>In a regression model setting, the goal is to establish whether or not a relationship exists between a response variable and a set of predictor variables. Further, if a relationship does exist, the goal is then to be able to describe this relationship as best as possible. A main assumption in <a href="Linear_regression" title="Linear regression">linear regression</a> is constant variance or (homoscedasticity), meaning that different response variables have the same variance in their errors, at every predictor level. This assumption works well when the response variable and the predictor variable are jointly <a href="Normal_distribution" title="Normal distribution">normal</a>. As we will see later, the variance function in the Normal setting is constant; however, we must find a way to quantify heteroscedasticity (non-constant variance) in the absence of joint Normality.
</p><p>When it is likely that the response follows a distribution that is a member of the exponential family, a <a href="Generalized_linear_model" title="Generalized linear model">generalized linear model</a> may be more appropriate to use, and moreover, when we wish not to force a parametric model onto our data, a <a href="Non-parametric_regression" class="mw-redirect" title="Non-parametric regression">non-parametric regression</a> approach can be useful. The importance of being able to model the variance as a function of the mean lies in improved inference (in a parametric setting), and estimation of the regression function in general, for any setting.
</p><p>Variance functions play a very important role in parameter estimation and inference. In general, maximum likelihood estimation requires that a likelihood function be defined. This requirement then implies that one must first specify the distribution of the response variables observed. However, to define a quasi-likelihood, one need only specify a relationship between the mean and the variance of the observations to then be able to use the quasi-likelihood function for estimation.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="Quasi-likelihood" title="Quasi-likelihood">Quasi-likelihood</a> estimation is particularly useful when there is <a href="Overdispersion" title="Overdispersion">overdispersion</a>. Overdispersion occurs when there is more variability in the data than there should otherwise be expected according to the assumed distribution of the data.
</p><p>In summary, to ensure efficient inference of the regression parameters and the regression function, the heteroscedasticity must be accounted for. Variance functions quantify the relationship between the variance and the mean of the observed data and hence play a significant role in regression estimation and inference.
</p>
<div class="mw-heading mw-heading2"><h2 id="Types">Types</h2></div>
<p>The variance function and its applications come up in many areas of statistical analysis. A very important use of this function is in the framework of <a href="Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a> and <a href="Non-parametric_regression" class="mw-redirect" title="Non-parametric regression">non-parametric regression</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Generalized_linear_model">Generalized linear model</h3></div>
<p>When a member of the <a href="Exponential_family" title="Exponential family">exponential family</a> has been specified, the variance function can easily be derived.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 29">: 29 </span></sup> The general form of the variance function is presented under the exponential family context, as well as specific forms for Normal, Bernoulli, Poisson, and Gamma. In addition, we describe the applications and use of variance functions in maximum likelihood estimation and quasi-likelihood estimation.
</p>
<div class="mw-heading mw-heading4"><h4 id="Derivation">Derivation</h4></div>
<p>The <b>generalized linear model (GLM)</b>, is a generalization of ordinary regression analysis that extends to any member of the <a href="Exponential_family" title="Exponential family">exponential family</a>. It is particularly useful when the response variable is categorical, binary or subject to a constraint (e.g. only positive responses make sense). A quick summary of the components of a GLM are summarized on this page, but for more details and information see the page on <a href="Generalized_linear_models" class="mw-redirect" title="Generalized linear models">generalized linear models</a>.
</p><p>A <b>GLM</b> consists of three main ingredients:
</p>
<dl><dd>1. Random Component: a distribution of <b>y</b> from the exponential family, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[y\mid X]=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E[y\mid X]=\mu }</annotation>
</semantics>
</math></span><img src="./07291ee76d720db9030e7ef20aaa874fb239676e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.642ex; height:2.843ex;" alt="{\displaystyle E[y\mid X]=\mu }" loading="lazy"></span></dd>
<dd>2. Linear predictor: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =XB=\sum _{j=1}^{p}X_{ij}^{T}B_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>η<!-- η --></mi>
<mo>=</mo>
<mi>X</mi>
<mi>B</mi>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</munderover>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msubsup>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta =XB=\sum _{j=1}^{p}X_{ij}^{T}B_{j}}</annotation>
</semantics>
</math></span><img src="./c739f036d79e77f40adb54b37e092be86bda8aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.928ex; height:7.343ex;" alt="{\displaystyle \eta =XB=\sum _{j=1}^{p}X_{ij}^{T}B_{j}}" loading="lazy"></span></dd>
<dd>3. Link function: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =g(\mu ),\mu =g^{-1}(\eta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>η<!-- η --></mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>η<!-- η --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta =g(\mu ),\mu =g^{-1}(\eta )}</annotation>
</semantics>
</math></span><img src="./86cea0a03bb8094ecaf61da1d10ef1469b5f4925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.559ex; height:3.176ex;" alt="{\displaystyle \eta =g(\mu ),\mu =g^{-1}(\eta )}" loading="lazy"></span></dd></dl>
<p>First it is important to derive a couple key properties of the exponential family.
</p><p>Any random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textit {y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext class="MJX-tex-mathit" mathvariant="italic">y</mtext>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\textit {y}}}</annotation>
</semantics>
</math></span><img src="./d12e3512d6a85cb15694775970937b392f6c6248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.084ex; width:1.214ex; height:2.009ex;" alt="{\displaystyle {\textit {y}}}" loading="lazy"></span> in the exponential family has a probability density function of the form,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y,\theta ,\phi )=\exp \left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(y,\theta ,\phi )=\exp \left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)}</annotation>
</semantics>
</math></span><img src="./ec7312b2894e19b6e22c5a469275fdcd9e7aa8f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.91ex; height:6.343ex;" alt="{\displaystyle f(y,\theta ,\phi )=\exp \left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)}" loading="lazy"></span></dd></dl>
<p>with loglikelihood,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (\theta ,y,\phi )=\log(f(y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ℓ<!-- ℓ --></mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ell (\theta ,y,\phi )=\log(f(y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}</annotation>
</semantics>
</math></span><img src="./9682834c5152d9be0acfd22c46734d4337e990b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.294ex; height:6.176ex;" alt="{\displaystyle \ell (\theta ,y,\phi )=\log(f(y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}" loading="lazy"></span></dd></dl>
<p>Here, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> is the canonical parameter and the parameter of interest, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> is a nuisance parameter which plays a role in the variance.
We use the <b>Bartlett's Identities</b> to derive a general expression for the <b>variance function</b>.
The first and second Bartlett results ensures that under suitable conditions (see <a href="Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a>), for a density function dependent on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,f_{\theta }()}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta ,f_{\theta }()}</annotation>
</semantics>
</math></span><img src="./44f0b4222f8f5b44a63916b903d68d239d4be9d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.076ex; height:2.843ex;" alt="{\displaystyle \theta ,f_{\theta }()}" loading="lazy"></span>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]=0}</annotation>
</semantics>
</math></span><img src="./f92aa025c1f0e0e2b5ec5fdd309cf9136cc3e8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.822ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]=0}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\log(f_{\theta }(y))\right]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>+</mo>
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\log(f_{\theta }(y))\right]=0}</annotation>
</semantics>
</math></span><img src="./3022dafdd32dfe00aabc581ea809bed35c43a6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.512ex; height:6.343ex;" alt="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\log(f_{\theta }(y))\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\log(f_{\theta }(y))\right]=0}" loading="lazy"></span></dd></dl>
<p>These identities lead to simple calculations of the expected value and variance of any random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textit {y}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext class="MJX-tex-mathit" mathvariant="italic">y</mtext>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\textit {y}}}</annotation>
</semantics>
</math></span><img src="./d12e3512d6a85cb15694775970937b392f6c6248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.084ex; width:1.214ex; height:2.009ex;" alt="{\displaystyle {\textit {y}}}" loading="lazy"></span> in the exponential family <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\theta }[y],Var_{\theta }[y]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
<mo>,</mo>
<mi>V</mi>
<mi>a</mi>
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{\theta }[y],Var_{\theta }[y]}</annotation>
</semantics>
</math></span><img src="./c568a80c4c2e771b25bd6cd95cf5ad7078fefa81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.72ex; height:2.843ex;" alt="{\displaystyle E_{\theta }[y],Var_{\theta }[y]}" loading="lazy"></span>.
</p><p><b>Expected value of <i>Y</i>:</b>
Taking the first derivative with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> of the log of the density in the exponential family form described above, we have
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial \theta }}\log(f(y,\theta ,\phi ))={\frac {\partial }{\partial \theta }}\left[{\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right]={\frac {y-b'(\theta )}{\phi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial \theta }}\log(f(y,\theta ,\phi ))={\frac {\partial }{\partial \theta }}\left[{\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right]={\frac {y-b'(\theta )}{\phi }}}</annotation>
</semantics>
</math></span><img src="./f1c5ac99a1a0656d5eee5823778e00fb34ba3aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.949ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial }{\partial \theta }}\log(f(y,\theta ,\phi ))={\frac {\partial }{\partial \theta }}\left[{\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right]={\frac {y-b'(\theta )}{\phi }}}" loading="lazy"></span></dd></dl>
<p>Then taking the expected value and setting it equal to zero leads to,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]={\frac {\operatorname {E} _{\theta }[y]-b'(\theta )}{\phi }}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]={\frac {\operatorname {E} _{\theta }[y]-b'(\theta )}{\phi }}=0}</annotation>
</semantics>
</math></span><img src="./e373e73c19e0de5b79c5de0a4dcd47de9cb6e162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.108ex; height:6.343ex;" alt="{\displaystyle \operatorname {E} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]={\frac {\operatorname {E} _{\theta }[y]-b'(\theta )}{\phi }}=0}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{\theta }[y]=b'(\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }[y]=b'(\theta )}</annotation>
</semantics>
</math></span><img src="./b345dfb730473bbda32a0e52c48ef1aac55049b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.716ex; height:3.009ex;" alt="{\displaystyle \operatorname {E} _{\theta }[y]=b'(\theta )}" loading="lazy"></span></dd></dl>
<p><b>Variance of Y:</b>
To compute the variance we use the second Bartlett identity,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>+</mo>
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]=0}</annotation>
</semantics>
</math></span><img src="./cbcb190685e4e10f56b4fc850471c16d3d01fd46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.68ex; height:6.343ex;" alt="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {\partial }{\partial \theta }}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]+\operatorname {E} _{\theta }\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\left({\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )\right)\right]=0}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]+\operatorname {E} _{\theta }\left[{\frac {-b''(\theta )}{\phi }}\right]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mo>+</mo>
<msub>
<mi mathvariant="normal">E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]+\operatorname {E} _{\theta }\left[{\frac {-b''(\theta )}{\phi }}\right]=0}</annotation>
</semantics>
</math></span><img src="./33fdd03408d8f4923faa09d99ed071fa0d4c32b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.511ex; height:6.343ex;" alt="{\displaystyle \operatorname {Var} _{\theta }\left[{\frac {y-b'(\theta )}{\phi }}\right]+\operatorname {E} _{\theta }\left[{\frac {-b''(\theta )}{\phi }}\right]=0}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }\left[y\right]=b''(\theta )\phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mi>y</mi>
<mo>]</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }\left[y\right]=b''(\theta )\phi }</annotation>
</semantics>
</math></span><img src="./6f24549fbfcd7b04ba21bf84e087568bc0fb3e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.788ex; height:3.009ex;" alt="{\displaystyle \operatorname {Var} _{\theta }\left[y\right]=b''(\theta )\phi }" loading="lazy"></span></dd></dl>
<p>We have now a relationship between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span>, namely
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =b'(\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =b'(\theta )}</annotation>
</semantics>
</math></span><img src="./40da40925e781b84d72faea802e6772d244dfdd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.082ex; height:3.009ex;" alt="{\displaystyle \mu =b'(\theta )}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =b'^{-1}(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<msup>
<mi>b</mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =b'^{-1}(\mu )}</annotation>
</semantics>
</math></span><img src="./0df98c239b0140f8019a3d6ff5422815c265685d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.183ex; height:3.176ex;" alt="{\displaystyle \theta =b'^{-1}(\mu )}" loading="lazy"></span>, which allows for a relationship between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> and the variance,</dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\theta )=b''(\theta )={\text{the part of the variance that depends on }}\theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>the part of the variance that depends on&nbsp;</mtext>
</mrow>
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\theta )=b''(\theta )={\text{the part of the variance that depends on }}\theta }</annotation>
</semantics>
</math></span><img src="./939b407e2c40c2a35b0423f72c3c14a109e54f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.597ex; height:3.009ex;" alt="{\displaystyle V(\theta )=b''(\theta )={\text{the part of the variance that depends on }}\theta }" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {V} (\mu )=b''(b'^{-1}(\mu )).\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">V</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mi>b</mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {V} (\mu )=b''(b'^{-1}(\mu )).\,}</annotation>
</semantics>
</math></span><img src="./b7068db2fc25076eca6a90e87b099183f1ef7b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.024ex; height:3.176ex;" alt="{\displaystyle \operatorname {V} (\mu )=b''(b'^{-1}(\mu )).\,}" loading="lazy"></span></dd></dl>
<p>Note that because <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }\left[y\right]>0,b''(\theta )>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mi>y</mi>
<mo>]</mo>
</mrow>
<mo>&gt;</mo>
<mn>0</mn>
<mo>,</mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }\left[y\right]&gt;0,b''(\theta )&gt;0}</annotation>
</semantics>
</math></span><img src="./cffcd8970188beb3d0b3ed58fb36a4d863766cdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.86ex; height:3.009ex;" alt="{\displaystyle \operatorname {Var} _{\theta }\left[y\right]>0,b''(\theta )>0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b':\theta \rightarrow \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo>:</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b':\theta \rightarrow \mu }</annotation>
</semantics>
</math></span><img src="./d9293e11b90738661a82d917c412609ae77043d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.726ex; height:3.009ex;" alt="{\displaystyle b':\theta \rightarrow \mu }" loading="lazy"></span> is invertible.
We derive the variance function for a few common distributions.
</p>
<div class="mw-heading mw-heading4"><h4 id="Example_–_normal">Example – normal</h4></div>
<p>The <a href="Normal_distribution" title="Normal distribution">normal distribution</a> is a special case where the variance function is a constant. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim N(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>∼<!-- ∼ --></mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\sim N(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="./95b9676da6bb256ea09cd07acc669a2d1e452c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.947ex; height:3.176ex;" alt="{\displaystyle y\sim N(\mu ,\sigma ^{2})}" loading="lazy"></span> then we put the density function of <b>y</b> in the form of the exponential family described above:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)=\exp \left({\frac {y\mu -{\frac {\mu ^{2}}{2}}}{\sigma ^{2}}}-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>μ<!-- μ --></mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(y)=\exp \left({\frac {y\mu -{\frac {\mu ^{2}}{2}}}{\sigma ^{2}}}-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}\right)}</annotation>
</semantics>
</math></span><img src="./c8b6216a7b7ceed9850abd7b66ac565c0d639e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:43.509ex; height:9.509ex;" alt="{\displaystyle f(y)=\exp \left({\frac {y\mu -{\frac {\mu ^{2}}{2}}}{\sigma ^{2}}}-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}\right)}" loading="lazy"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\mu ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =\mu ,}</annotation>
</semantics>
</math></span><img src="./22ffd8517cdd996aff9247fa1bdbe9ebdf77d253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.237ex; height:2.676ex;" alt="{\displaystyle \theta =\mu ,}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(\theta )={\frac {\mu ^{2}}{2}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mn>2</mn>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b(\theta )={\frac {\mu ^{2}}{2}},}</annotation>
</semantics>
</math></span><img src="./f7b723e09e3bbf19b14ca9cd0b7e1ecd33c58e7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.935ex; height:5.676ex;" alt="{\displaystyle b(\theta )={\frac {\mu ^{2}}{2}},}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\sigma ^{2},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>=</mo>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi =\sigma ^{2},}</annotation>
</semantics>
</math></span><img src="./501d880b19a4a9e8881b80bc325a44784bb08cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.515ex; height:3.009ex;" alt="{\displaystyle \phi =\sigma ^{2},}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(y,\phi )=-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>π<!-- π --></mi>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c(y,\phi )=-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}}</annotation>
</semantics>
</math></span><img src="./13268e0462ee1bc295e7ac2a602cf0642ad05992.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.113ex; height:6.009ex;" alt="{\displaystyle c(y,\phi )=-{\frac {y^{2}}{2\sigma ^{2}}}-{\frac {1}{2}}\ln {2\pi \sigma ^{2}}}" loading="lazy"></span></dd></dl>
<p>To calculate the variance function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\mu )}</annotation>
</semantics>
</math></span><img src="./4026aa7cc24d6ca14a49b3e6529a6d22b8f26eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle V(\mu )}" loading="lazy"></span>, we first express <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> as a function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>. Then we transform <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\theta )}</annotation>
</semantics>
</math></span><img src="./5e77dc65a2239817d8f7c89b67ca8778a9f050ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.687ex; height:2.843ex;" alt="{\displaystyle V(\theta )}" loading="lazy"></span> into a function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =\mu }</annotation>
</semantics>
</math></span><img src="./ac9063d0c72c38a12998c729fb596f1ad5233a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle \theta =\mu }" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b'(\theta )=\theta =\operatorname {E} [y]=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b'(\theta )=\theta =\operatorname {E} [y]=\mu }</annotation>
</semantics>
</math></span><img src="./e20970e33b0dc581298f24b51a44e60857002f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.402ex; height:3.009ex;" alt="{\displaystyle b'(\theta )=\theta =\operatorname {E} [y]=\mu }" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\theta )=b''(\theta )=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\theta )=b''(\theta )=1}</annotation>
</semantics>
</math></span><img src="./1d7278f465476bf16784a4541912fe7884aa4c7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.081ex; height:3.009ex;" alt="{\displaystyle V(\theta )=b''(\theta )=1}" loading="lazy"></span></dd></dl>
<p>Therefore, the variance function is constant.
</p>
<div class="mw-heading mw-heading4"><h4 id="Example_–_Bernoulli">Example – Bernoulli</h4></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim {\text{Bernoulli}}(p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>∼<!-- ∼ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Bernoulli</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\sim {\text{Bernoulli}}(p)}</annotation>
</semantics>
</math></span><img src="./194f73d5742391bf99bee5a884fe5395405ce941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.51ex; height:2.843ex;" alt="{\displaystyle y\sim {\text{Bernoulli}}(p)}" loading="lazy"></span>, then we express the density of the <a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a> in exponential family form,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)=\exp \left(y\ln {\frac {p}{1-p}}+\ln(1-p)\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>p</mi>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(y)=\exp \left(y\ln {\frac {p}{1-p}}+\ln(1-p)\right)}</annotation>
</semantics>
</math></span><img src="./a80046cdf79591e7ecf3e7f331606d1c50e48102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.954ex; height:6.176ex;" alt="{\displaystyle f(y)=\exp \left(y\ln {\frac {p}{1-p}}+\ln(1-p)\right)}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\ln {\frac {p}{1-p}}=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>p</mi>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>p</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =\ln {\frac {p}{1-p}}=}</annotation>
</semantics>
</math></span><img src="./ad3e25b03a3dde0c11d19577ebfae976ff8d88fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.977ex; height:5.343ex;" alt="{\displaystyle \theta =\ln {\frac {p}{1-p}}=}" loading="lazy"></span> <a href="Logit" title="Logit">logit</a>(p), which gives us <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {e^{\theta }}{1+e^{\theta }}}=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p={\frac {e^{\theta }}{1+e^{\theta }}}=}</annotation>
</semantics>
</math></span><img src="./6e7068ef064ce8ccd33590a2b01fdc0967eabc4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:13.736ex; height:6.176ex;" alt="{\displaystyle p={\frac {e^{\theta }}{1+e^{\theta }}}=}" loading="lazy"></span> <a href="Expit" class="mw-redirect" title="Expit">expit</a><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\theta )}</annotation>
</semantics>
</math></span><img src="./439930e0197465b29c7efcb17deaab3313303710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.9ex; height:2.843ex;" alt="{\displaystyle (\theta )}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(\theta )=\ln(1+e^{\theta })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b(\theta )=\ln(1+e^{\theta })}</annotation>
</semantics>
</math></span><img src="./140ba5615f08502411d346164034a4153a78a3f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.834ex; height:3.176ex;" alt="{\displaystyle b(\theta )=\ln(1+e^{\theta })}" loading="lazy"></span> and</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b'(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b'(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}=}</annotation>
</semantics>
</math></span><img src="./7b75e4b60ee139a80c365b450e6f69748f96a39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.06ex; height:6.176ex;" alt="{\displaystyle b'(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}=}" loading="lazy"></span> <a href="Expit" class="mw-redirect" title="Expit">expit</a><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta )=p=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>p</mi>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\theta )=p=\mu }</annotation>
</semantics>
</math></span><img src="./830179e166fa1ade1e2922b94b4f07cfcedfb0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.668ex; height:2.843ex;" alt="{\displaystyle (\theta )=p=\mu }" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b''(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}-\left({\frac {e^{\theta }}{1+e^{\theta }}}\right)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b''(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}-\left({\frac {e^{\theta }}{1+e^{\theta }}}\right)^{2}}</annotation>
</semantics>
</math></span><img src="./2e63975d3a5a34b8d308124cc24d6687c79ce11e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.3ex; height:6.843ex;" alt="{\displaystyle b''(\theta )={\frac {e^{\theta }}{1+e^{\theta }}}-\left({\frac {e^{\theta }}{1+e^{\theta }}}\right)^{2}}" loading="lazy"></span></dd></dl>
<p>This give us
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )=\mu (1-\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\mu )=\mu (1-\mu )}</annotation>
</semantics>
</math></span><img src="./ccbb36e301fc89367f479e9f85df652906a71631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.712ex; height:2.843ex;" alt="{\displaystyle V(\mu )=\mu (1-\mu )}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading4"><h4 id="Example_–_Poisson">Example – Poisson</h4></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim {\text{Poisson}}(\lambda )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>∼<!-- ∼ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>Poisson</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\sim {\text{Poisson}}(\lambda )}</annotation>
</semantics>
</math></span><img src="./4cc30ff08665599c9255169f727c05cff35559f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.098ex; height:2.843ex;" alt="{\displaystyle y\sim {\text{Poisson}}(\lambda )}" loading="lazy"></span>, then we express the density of the <a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a> in exponential family form,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)=\exp(y\ln \lambda -\ln \lambda )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>λ<!-- λ --></mi>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>λ<!-- λ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(y)=\exp(y\ln \lambda -\ln \lambda )}</annotation>
</semantics>
</math></span><img src="./88ba12c50c51693218812a3b7cf787d9becd48ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.45ex; height:2.843ex;" alt="{\displaystyle f(y)=\exp(y\ln \lambda -\ln \lambda )}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\ln \lambda =}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>λ<!-- λ --></mi>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta =\ln \lambda =}</annotation>
</semantics>
</math></span><img src="./c801290fa20024c3f7bc7b07711f63323fec5fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.324ex; height:2.176ex;" alt="{\displaystyle \theta =\ln \lambda =}" loading="lazy"></span> which gives us <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =e^{\theta }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda =e^{\theta }}</annotation>
</semantics>
</math></span><img src="./502aaee506dde63c8dc044c4c3cfe3e4689270e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.541ex; height:2.676ex;" alt="{\displaystyle \lambda =e^{\theta }}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(\theta )=e^{\theta }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b(\theta )=e^{\theta }}</annotation>
</semantics>
</math></span><img src="./4f09ca99546d136a3cf3111557330de305fcda75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.083ex; height:3.176ex;" alt="{\displaystyle b(\theta )=e^{\theta }}" loading="lazy"></span> and</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b'(\theta )=e^{\theta }=\lambda =\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b'(\theta )=e^{\theta }=\lambda =\mu }</annotation>
</semantics>
</math></span><img src="./23e84d93421a215e6a6de0c2844460f659029527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.721ex; height:3.176ex;" alt="{\displaystyle b'(\theta )=e^{\theta }=\lambda =\mu }" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b''(\theta )=e^{\theta }=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msup>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b''(\theta )=e^{\theta }=\mu }</annotation>
</semantics>
</math></span><img src="./50014b632bc6ec3d9ac31f791bd8394fadf9b915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.72ex; height:3.176ex;" alt="{\displaystyle b''(\theta )=e^{\theta }=\mu }" loading="lazy"></span></dd></dl>
<p>This give us
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\mu )=\mu }</annotation>
</semantics>
</math></span><img src="./c15a2d317e8045d29ca91d11f232c8e71d268170.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.498ex; height:2.843ex;" alt="{\displaystyle V(\mu )=\mu }" loading="lazy"></span></dd></dl>
<p>Here we see the central property of Poisson data, that the variance is equal to the mean.
</p>
<div class="mw-heading mw-heading4"><h4 id="Example_–_Gamma">Example – Gamma</h4></div>
<p>The <a href="Gamma_distribution" title="Gamma distribution">Gamma distribution</a> and density function can be expressed under different parametrizations. We will use the form of the gamma with parameters <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu ,\nu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\mu ,\nu )}</annotation>
</semantics>
</math></span><img src="./ea1a07b0389d3f118fb3e2e81473a105c9c70705.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.477ex; height:2.843ex;" alt="{\displaystyle (\mu ,\nu )}" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mu ,\nu }(y)={\frac {1}{\Gamma (\nu )y}}\left({\frac {\nu y}{\mu }}\right)^{\nu }e^{-{\frac {\nu y}{\mu }}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">)</mo>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>ν<!-- ν --></mi>
<mi>y</mi>
</mrow>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>ν<!-- ν --></mi>
<mi>y</mi>
</mrow>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\mu ,\nu }(y)={\frac {1}{\Gamma (\nu )y}}\left({\frac {\nu y}{\mu }}\right)^{\nu }e^{-{\frac {\nu y}{\mu }}}}</annotation>
</semantics>
</math></span><img src="./694f6fef5d9103649bc26689c434cca332521546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.79ex; height:6.343ex;" alt="{\displaystyle f_{\mu ,\nu }(y)={\frac {1}{\Gamma (\nu )y}}\left({\frac {\nu y}{\mu }}\right)^{\nu }e^{-{\frac {\nu y}{\mu }}}}" loading="lazy"></span></dd></dl>
<p>Then in exponential family form we have
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mu ,\nu }(y)=\exp \left({\frac {-{\frac {1}{\mu }}y+\ln({\frac {1}{\mu }})}{\frac {1}{\nu }}}+\ln \left({\frac {\nu ^{\nu }y^{\nu -1}}{\Gamma (\nu )}}\right)\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>ν<!-- ν --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mi>y</mi>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mfrac>
</mrow>
<mo>+</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>ν<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{\mu ,\nu }(y)=\exp \left({\frac {-{\frac {1}{\mu }}y+\ln({\frac {1}{\mu }})}{\frac {1}{\nu }}}+\ln \left({\frac {\nu ^{\nu }y^{\nu -1}}{\Gamma (\nu )}}\right)\right)}</annotation>
</semantics>
</math></span><img src="./f18da08920cf9d8f5b818ebe074922688dd00625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:47.18ex; height:8.176ex;" alt="{\displaystyle f_{\mu ,\nu }(y)=\exp \left({\frac {-{\frac {1}{\mu }}y+\ln({\frac {1}{\mu }})}{\frac {1}{\nu }}}+\ln \left({\frac {\nu ^{\nu }y^{\nu -1}}{\Gamma (\nu )}}\right)\right)}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {-1}{\mu }}\rightarrow \mu ={\frac {-1}{\theta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mi>θ<!-- θ --></mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {-1}{\mu }}\rightarrow \mu ={\frac {-1}{\theta }}}</annotation>
</semantics>
</math></span><img src="./4be11c0f7e30331da81eb8965c3aaa61a62441b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.917ex; height:5.676ex;" alt="{\displaystyle \theta ={\frac {-1}{\mu }}\rightarrow \mu ={\frac {-1}{\theta }}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ={\frac {1}{\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi ={\frac {1}{\nu }}}</annotation>
</semantics>
</math></span><img src="./1493beeb9db5a1bff11dcc6513295226670805ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.552ex; height:5.176ex;" alt="{\displaystyle \phi ={\frac {1}{\nu }}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(\theta )=-\ln(-\theta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b(\theta )=-\ln(-\theta )}</annotation>
</semantics>
</math></span><img src="./8d5bc6b7fa4452d2ee1f1ebee0abf7a4dce28f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.838ex; height:2.843ex;" alt="{\displaystyle b(\theta )=-\ln(-\theta )}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b'(\theta )={\frac {-1}{\theta }}={\frac {-1}{\frac {-1}{\mu }}}=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mi>θ<!-- θ --></mi>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mi>μ<!-- μ --></mi>
</mfrac>
</mfrac>
</mrow>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b'(\theta )={\frac {-1}{\theta }}={\frac {-1}{\frac {-1}{\mu }}}=\mu }</annotation>
</semantics>
</math></span><img src="./5b8ff7d98e6a724d6d98bb62f9e857072ab95dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:22.893ex; height:7.009ex;" alt="{\displaystyle b'(\theta )={\frac {-1}{\theta }}={\frac {-1}{\frac {-1}{\mu }}}=\mu }" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b''(\theta )={\frac {1}{\theta ^{2}}}=\mu ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>=</mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b''(\theta )={\frac {1}{\theta ^{2}}}=\mu ^{2}}</annotation>
</semantics>
</math></span><img src="./5fdc2b997a428485250403816570b4b15a052ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.668ex; height:5.509ex;" alt="{\displaystyle b''(\theta )={\frac {1}{\theta ^{2}}}=\mu ^{2}}" loading="lazy"></span></dd></dl>
<p>And we have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )=\mu ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\mu )=\mu ^{2}}</annotation>
</semantics>
</math></span><img src="./52ae672cc8dcdc8c422f3123849fc0d7b72c90ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.553ex; height:3.176ex;" alt="{\displaystyle V(\mu )=\mu ^{2}}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading4"><h4 id="Application_–_weighted_least_squares">Application – weighted least squares</h4></div>
<p>A very important application of the variance function is its use in parameter estimation and inference when the response variable is of the required exponential family form as well as in some cases when it is not (which we will discuss in <a href="Quasi-likelihood" title="Quasi-likelihood">quasi-likelihood</a>). Weighted <a href="Least_squares" title="Least squares">least squares</a> (WLS) is a special case of generalized least squares. Each term in the WLS criterion includes a weight that determines that the influence each observation has on the final parameter estimates. As in regular least squares, the goal is to estimate the unknown parameters in the regression function by finding values for parameter estimates that minimize the sum of the squared deviations between the observed responses and the functional portion of the model.
</p><p>While WLS assumes independence of observations it does not assume equal variance and is therefore a solution for parameter estimation in the presence of heteroscedasticity. The <a href="Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">Gauss–Markov theorem</a> and <a href="Alexander_Aitken" title="Alexander Aitken">Aitken</a> demonstrate that the <a href="Best_linear_unbiased_estimator" class="mw-redirect" title="Best linear unbiased estimator">best linear unbiased estimator</a> (BLUE), the unbiased estimator with minimum variance, has each weight equal to the reciprocal of the variance of the measurement.
</p><p>In the GLM framework, our goal is to estimate parameters <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=g(E[y\mid X])=X\beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>X</mi>
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z=g(E[y\mid X])=X\beta }</annotation>
</semantics>
</math></span><img src="./e36bc55cabcb5f79fd32d81aba1e12eb2e81625d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.257ex; height:2.843ex;" alt="{\displaystyle Z=g(E[y\mid X])=X\beta }" loading="lazy"></span>. Therefore, we would like to minimize <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Z-XB)^{T}W(Z-XB)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>Z</mi>
<mo>−<!-- − --></mo>
<mi>X</mi>
<mi>B</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>Z</mi>
<mo>−<!-- − --></mo>
<mi>X</mi>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (Z-XB)^{T}W(Z-XB)}</annotation>
</semantics>
</math></span><img src="./4d8a46bda4e9d1837ff3d6183011ba03c005abf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.973ex; height:3.176ex;" alt="{\displaystyle (Z-XB)^{T}W(Z-XB)}" loading="lazy"></span> and if we define the weight matrix <b>W</b> as
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {W} _{n\times n}={\begin{bmatrix}{\frac {1}{\phi V(\mu _{1})g'(\mu _{1})^{2}}}&amp;0&amp;\cdots &amp;0&amp;0\\0&amp;{\frac {1}{\phi V(\mu _{2})g'(\mu _{2})^{2}}}&amp;0&amp;\cdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\0&amp;\cdots &amp;\cdots &amp;0&amp;{\frac {1}{\phi V(\mu _{n})g'(\mu _{n})^{2}}}\end{bmatrix}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mi>W</mi>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</munder>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \underbrace {W} _{n\times n}={\begin{bmatrix}{\frac {1}{\phi V(\mu _{1})g'(\mu _{1})^{2}}}&amp;0&amp;\cdots &amp;0&amp;0\\0&amp;{\frac {1}{\phi V(\mu _{2})g'(\mu _{2})^{2}}}&amp;0&amp;\cdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\0&amp;\cdots &amp;\cdots &amp;0&amp;{\frac {1}{\phi V(\mu _{n})g'(\mu _{n})^{2}}}\end{bmatrix}},}</annotation>
</semantics>
</math></span><img src="./ea884bff785598f59c0790fa993c0d5a2284d13b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; margin-left: -0.157ex; width:60.765ex; height:23.176ex;" alt="{\displaystyle \underbrace {W} _{n\times n}={\begin{bmatrix}{\frac {1}{\phi V(\mu _{1})g'(\mu _{1})^{2}}}&amp;0&amp;\cdots &amp;0&amp;0\\0&amp;{\frac {1}{\phi V(\mu _{2})g'(\mu _{2})^{2}}}&amp;0&amp;\cdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;0\\0&amp;\cdots &amp;\cdots &amp;0&amp;{\frac {1}{\phi V(\mu _{n})g'(\mu _{n})^{2}}}\end{bmatrix}},}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ,V(\mu ),g(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi ,V(\mu ),g(\mu )}</annotation>
</semantics>
</math></span><img src="./38780bbbc90df0bac43ad71829783a50888e96f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.778ex; height:2.843ex;" alt="{\displaystyle \phi ,V(\mu ),g(\mu )}" loading="lazy"></span> are defined in the previous section, it allows for <a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">iteratively reweighted least squares</a> (IRLS) estimation of the parameters. See the section on <a href="Iteratively_reweighted_least_squares" title="Iteratively reweighted least squares">iteratively reweighted least squares</a> for more derivation and information.
</p><p>Also, important to note is that when the weight matrix is of the form described here, minimizing the expression <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Z-XB)^{T}W(Z-XB)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>Z</mi>
<mo>−<!-- − --></mo>
<mi>X</mi>
<mi>B</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>Z</mi>
<mo>−<!-- − --></mo>
<mi>X</mi>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (Z-XB)^{T}W(Z-XB)}</annotation>
</semantics>
</math></span><img src="./4d8a46bda4e9d1837ff3d6183011ba03c005abf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.973ex; height:3.176ex;" alt="{\displaystyle (Z-XB)^{T}W(Z-XB)}" loading="lazy"></span> also minimizes the Pearson distance. See <a href="Distance_correlation" title="Distance correlation">Distance correlation</a> for more.
</p><p>The matrix <b>W</b> falls right out of the estimating equations for estimation of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>. Maximum likelihood estimation for each parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{r},1\leq r\leq p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
<mo>,</mo>
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>r</mi>
<mo>≤<!-- ≤ --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta _{r},1\leq r\leq p}</annotation>
</semantics>
</math></span><img src="./27e89a9c36af10d1622158eb0951d96ff028df22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.901ex; height:2.509ex;" alt="{\displaystyle \beta _{r},1\leq r\leq p}" loading="lazy"></span>, requires
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{\frac {\partial l_{i}}{\partial \beta _{r}}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>l</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{\frac {\partial l_{i}}{\partial \beta _{r}}}=0}</annotation>
</semantics>
</math></span><img src="./7990fb1fccb1023ce3017bbb875f4b4dc8e7ed9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.447ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}{\frac {\partial l_{i}}{\partial \beta _{r}}}=0}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {l} (\theta ,y,\phi )=\log(\operatorname {f} (y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">l</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">f</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mi>b</mi>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mi>c</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {l} (\theta ,y,\phi )=\log(\operatorname {f} (y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}</annotation>
</semantics>
</math></span><img src="./513639bb6fdda38ad5f629801f97f0bb541605b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.558ex; height:6.176ex;" alt="{\displaystyle \operatorname {l} (\theta ,y,\phi )=\log(\operatorname {f} (y,\theta ,\phi ))={\frac {y\theta -b(\theta )}{\phi }}-c(y,\phi )}" loading="lazy"></span> is the log-likelihood.</dd></dl>
<p>Looking at a single observation we have,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {\partial l}{\partial \theta }}{\frac {\partial \theta }{\partial \mu }}{\frac {\partial \mu }{\partial \eta }}{\frac {\partial \eta }{\partial \beta _{r}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {\partial l}{\partial \theta }}{\frac {\partial \theta }{\partial \mu }}{\frac {\partial \mu }{\partial \eta }}{\frac {\partial \eta }{\partial \beta _{r}}}}</annotation>
</semantics>
</math></span><img src="./e7839b92bba87232b4a744ae74860179099771dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.342ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {\partial l}{\partial \theta }}{\frac {\partial \theta }{\partial \mu }}{\frac {\partial \mu }{\partial \eta }}{\frac {\partial \eta }{\partial \beta _{r}}}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \eta }{\partial \beta _{r}}}=x_{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \eta }{\partial \beta _{r}}}=x_{r}}</annotation>
</semantics>
</math></span><img src="./4eeafac775797aa15fd84a1e14964dca0a9c0387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.846ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \eta }{\partial \beta _{r}}}=x_{r}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial l}{\partial \theta }}={\frac {y-b'(\theta )}{\phi }}={\frac {y-\mu }{\phi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial l}{\partial \theta }}={\frac {y-b'(\theta )}{\phi }}={\frac {y-\mu }{\phi }}}</annotation>
</semantics>
</math></span><img src="./89c06e9effa392358c8eaa9990eca1097ce690b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.089ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial l}{\partial \theta }}={\frac {y-b'(\theta )}{\phi }}={\frac {y-\mu }{\phi }}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \theta }{\partial \mu }}={\frac {\partial b'^{-1}(\mu )}{\mu }}={\frac {1}{b''(b'(\mu ))}}={\frac {1}{V(\mu )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>θ<!-- θ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msup>
<mi>b</mi>
<mrow>
<mo class="MJX-variant">′</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mi>b</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mi>b</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \theta }{\partial \mu }}={\frac {\partial b'^{-1}(\mu )}{\mu }}={\frac {1}{b''(b'(\mu ))}}={\frac {1}{V(\mu )}}}</annotation>
</semantics>
</math></span><img src="./a1d8feb4c29cab03e0125aa8b84bb5aedec0d7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.507ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \theta }{\partial \mu }}={\frac {\partial b'^{-1}(\mu )}{\mu }}={\frac {1}{b''(b'(\mu ))}}={\frac {1}{V(\mu )}}}" loading="lazy"></span></dd></dl>
<p>This gives us
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {y-\mu }{\phi V(\mu )}}{\frac {\partial \mu }{\partial \eta }}x_{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {y-\mu }{\phi V(\mu )}}{\frac {\partial \mu }{\partial \eta }}x_{r}}</annotation>
</semantics>
</math></span><img src="./ffb4a18c6341823b90e24755c4c0b38e0e9aa624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.621ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}={\frac {y-\mu }{\phi V(\mu )}}{\frac {\partial \mu }{\partial \eta }}x_{r}}" loading="lazy"></span>, and noting that</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \eta }{\partial \mu }}=g'(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \eta }{\partial \mu }}=g'(\mu )}</annotation>
</semantics>
</math></span><img src="./d5da876687f9ab6a9d8cc627923114c52bd62706.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.668ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \eta }{\partial \mu }}=g'(\mu )}" loading="lazy"></span> we have that</dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}=(y-\mu )W{\frac {\partial \eta }{\partial \mu }}x_{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>η<!-- η --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}=(y-\mu )W{\frac {\partial \eta }{\partial \mu }}x_{r}}</annotation>
</semantics>
</math></span><img src="./bbe3250c7a26c080e60d3f1489faa0d0783afb31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.044ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial l}{\partial \beta _{r}}}=(y-\mu )W{\frac {\partial \eta }{\partial \mu }}x_{r}}" loading="lazy"></span></dd></dl>
<p>The Hessian matrix is determined in a similar manner and can be shown to be,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=X^{T}(y-\mu )\left[{\frac {\partial }{\beta _{s}}}W{\frac {\partial }{\beta _{r}}}\right]-X^{T}WX}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mfrac>
</mrow>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>−<!-- − --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>W</mi>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H=X^{T}(y-\mu )\left[{\frac {\partial }{\beta _{s}}}W{\frac {\partial }{\beta _{r}}}\right]-X^{T}WX}</annotation>
</semantics>
</math></span><img src="./ed90445d01c49c74c8a827cf91478dc1e6744762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.955ex; height:6.176ex;" alt="{\displaystyle H=X^{T}(y-\mu )\left[{\frac {\partial }{\beta _{s}}}W{\frac {\partial }{\beta _{r}}}\right]-X^{T}WX}" loading="lazy"></span></dd></dl>
<p>Noticing that the Fisher Information (FI),
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{FI}}=-E[H]=X^{T}WX}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>FI</mtext>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>H</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>W</mi>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{FI}}=-E[H]=X^{T}WX}</annotation>
</semantics>
</math></span><img src="./e50dbb1186cdb2e0e421e5a5c10c1cb98f249712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.297ex; height:3.176ex;" alt="{\displaystyle {\text{FI}}=-E[H]=X^{T}WX}" loading="lazy"></span>, allows for asymptotic approximation of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}}</annotation>
</semantics>
</math></span><img src="./efdb50e00928e4013750a476dab75eeb3cbd5799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.451ex; height:3.176ex;" alt="{\displaystyle {\hat {\beta }}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\beta }}\sim N_{p}(\beta ,(X^{T}WX)^{-1})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>β<!-- β --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>∼<!-- ∼ --></mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo>,</mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<mi>W</mi>
<mi>X</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}\sim N_{p}(\beta ,(X^{T}WX)^{-1})}</annotation>
</semantics>
</math></span><img src="./8ca92d1ba91516dac4a025e79413c17fe4210bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.594ex; height:3.509ex;" alt="{\displaystyle {\hat {\beta }}\sim N_{p}(\beta ,(X^{T}WX)^{-1})}" loading="lazy"></span>, and hence inference can be performed.</dd></dl>
<div class="mw-heading mw-heading4"><h4 id="Application_–_quasi-likelihood">Application – quasi-likelihood</h4></div>
<p>Because most features of <b>GLMs</b> only depend on the first two moments of the distribution, rather than the entire distribution, the quasi-likelihood can be developed by just specifying a link function and a variance function. That is, we need to specify
</p>
<ul><li>the link function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[y]=\mu =g^{-1}(\eta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>η<!-- η --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E[y]=\mu =g^{-1}(\eta )}</annotation>
</semantics>
</math></span><img src="./640a9105699c4e4f010c65abcbbef873342b8343.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.253ex; height:3.176ex;" alt="{\displaystyle E[y]=\mu =g^{-1}(\eta )}" loading="lazy"></span></li>
<li>the variance function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(\mu )}</annotation>
</semantics>
</math></span><img src="./4026aa7cc24d6ca14a49b3e6529a6d22b8f26eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.998ex; height:2.843ex;" alt="{\displaystyle V(\mu )}" loading="lazy"></span>, where the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} _{\theta }(y)=\sigma ^{2}V(\mu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Var</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>θ<!-- θ --></mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} _{\theta }(y)=\sigma ^{2}V(\mu )}</annotation>
</semantics>
</math></span><img src="./75e772772235bb4ea0eeaa5012e387191772971a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.267ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} _{\theta }(y)=\sigma ^{2}V(\mu )}" loading="lazy"></span></li></ul>
<p>With a specified variance function and link function we can develop, as alternatives to the log-<a href="Likelihood_function" title="Likelihood function">likelihood function</a>, the <a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">score function</a>, and the <a href="Fisher_information" title="Fisher information">Fisher information</a>, a <b><a href="Quasi-likelihood" title="Quasi-likelihood">quasi-likelihood</a></b>, a <b>quasi-score</b>, and the <b>quasi-information</b>. This allows for full inference of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>.
</p><p><b>Quasi-likelihood (QL)</b>
</p><p>Though called a <a href="Quasi-likelihood" title="Quasi-likelihood">quasi-likelihood</a>, this is in fact a quasi-<b>log</b>-likelihood. The QL for one observation is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}(\mu _{i},y_{i})=\int _{y_{i}}^{\mu _{i}}{\frac {y_{i}-t}{\sigma ^{2}V(t)}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{i}(\mu _{i},y_{i})=\int _{y_{i}}^{\mu _{i}}{\frac {y_{i}-t}{\sigma ^{2}V(t)}}\,dt}</annotation>
</semantics>
</math></span><img src="./16f2b3459f7b4062cee014290e9d1ccaf59282f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.519ex; height:6.343ex;" alt="{\displaystyle Q_{i}(\mu _{i},y_{i})=\int _{y_{i}}^{\mu _{i}}{\frac {y_{i}-t}{\sigma ^{2}V(t)}}\,dt}" loading="lazy"></span></dd></dl>
<p>And therefore the QL for all <b>n</b> observations is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(\mu ,y)=\sum _{i=1}^{n}Q_{i}(\mu _{i},y_{i})=\sum _{i=1}^{n}\int _{y_{i}}^{\mu _{i}}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q(\mu ,y)=\sum _{i=1}^{n}Q_{i}(\mu _{i},y_{i})=\sum _{i=1}^{n}\int _{y_{i}}^{\mu _{i}}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}</annotation>
</semantics>
</math></span><img src="./67e8dac8220108ba278a7e01ce84b4413b35b074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.34ex; height:6.843ex;" alt="{\displaystyle Q(\mu ,y)=\sum _{i=1}^{n}Q_{i}(\mu _{i},y_{i})=\sum _{i=1}^{n}\int _{y_{i}}^{\mu _{i}}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}" loading="lazy"></span></dd></dl>
<p>From the <b>QL</b> we have the <b>quasi-score</b>
</p><p><b>Quasi-score (QS)</b>
</p><p>Recall the <a href="Score_(statistics)" class="mw-redirect" title="Score (statistics)">score function</a>, <b>U</b>, for data with log-likelihood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {l} (\mu \mid y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">l</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>∣<!-- ∣ --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {l} (\mu \mid y)}</annotation>
</semantics>
</math></span><img src="./a80f6ab4ba01dc3eedcd0723fb3e484cf5369ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.95ex; height:2.843ex;" alt="{\displaystyle \operatorname {l} (\mu \mid y)}" loading="lazy"></span> is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\frac {\partial l}{d\mu }}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>l</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U={\frac {\partial l}{d\mu }}.}</annotation>
</semantics>
</math></span><img src="./1a26db7a68c7ad5cae7ed4b74c17b5b626c2a4b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.982ex; height:5.843ex;" alt="{\displaystyle U={\frac {\partial l}{d\mu }}.}" loading="lazy"></span></dd></dl>
<p>We obtain the quasi-score in an identical manner,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}</annotation>
</semantics>
</math></span><img src="./59a447708e1a7ad227fd4b062ff8b954a5c866e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.1ex; height:6.009ex;" alt="{\displaystyle U={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}" loading="lazy"></span></dd></dl>
<p>Noting that, for one observation the score is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial Q}{\partial \mu }}={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Q</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial Q}{\partial \mu }}={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}</annotation>
</semantics>
</math></span><img src="./687f764b7d5c4d5e7fb96137b4248d00b0179f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.31ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial Q}{\partial \mu }}={\frac {y-\mu }{\sigma ^{2}V(\mu )}}}" loading="lazy"></span></dd></dl>
<p>The first two Bartlett equations are satisfied for the quasi-score, namely
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[U]=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo stretchy="false">[</mo>
<mi>U</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E[U]=0}</annotation>
</semantics>
</math></span><img src="./e8aa4aa9908d6e3161af517a98583f57d5d8df09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.113ex; height:2.843ex;" alt="{\displaystyle E[U]=0}" loading="lazy"></span></dd></dl>
<p>and
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} (U)+E\left[{\frac {\partial U}{\partial \mu }}\right]=0.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>U</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>E</mi>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>U</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>μ<!-- μ --></mi>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (U)+E\left[{\frac {\partial U}{\partial \mu }}\right]=0.}</annotation>
</semantics>
</math></span><img src="./4dbb6dc91a5077b6809b6c4a09a9254ed7fbd1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.963ex; height:6.176ex;" alt="{\displaystyle \operatorname {Cov} (U)+E\left[{\frac {\partial U}{\partial \mu }}\right]=0.}" loading="lazy"></span></dd></dl>
<p>In addition, the quasi-score is linear in <b>y</b>.
</p><p>Ultimately the goal is to find information about the parameters of interest <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>. Both the QS and the QL are actually functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>. Recall, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =g^{-1}(\eta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>η<!-- η --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =g^{-1}(\eta )}</annotation>
</semantics>
</math></span><img src="./04d2d315d981619e0bcfa682fbe970eaf8fb9cc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.93ex; height:3.176ex;" alt="{\displaystyle \mu =g^{-1}(\eta )}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =X\beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>η<!-- η --></mi>
<mo>=</mo>
<mi>X</mi>
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta =X\beta }</annotation>
</semantics>
</math></span><img src="./8e074091c51bacfacc508b750cf3e64fe022dc6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.58ex; height:2.676ex;" alt="{\displaystyle \eta =X\beta }" loading="lazy"></span>, therefore,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =g^{-1}(X\beta ).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =g^{-1}(X\beta ).}</annotation>
</semantics>
</math></span><img src="./19831eaeaa20536d2ac0df04a7fa3bb96fca2c05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.719ex; height:3.176ex;" alt="{\displaystyle \mu =g^{-1}(X\beta ).}" loading="lazy"></span></dd></dl>
<p><b>Quasi-information (QI)</b>
</p><p>The <b>quasi-information</b>, is similar to the <a href="Fisher_information" title="Fisher information">Fisher information</a>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{b}=-\operatorname {E} \left[{\frac {\partial U}{\partial \beta }}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>U</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>β<!-- β --></mi>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{b}=-\operatorname {E} \left[{\frac {\partial U}{\partial \beta }}\right]}</annotation>
</semantics>
</math></span><img src="./6f7aafa025efbdc09fd9598df87d1a6832ece801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.008ex; height:6.176ex;" alt="{\displaystyle i_{b}=-\operatorname {E} \left[{\frac {\partial U}{\partial \beta }}\right]}" loading="lazy"></span></dd></dl>
<p><b>QL, QS, QI as functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span></b>
</p><p>The QL, QS and QI all provide the building blocks for inference about the parameters of interest and therefore it is important to express the QL, QS and QI all as functions of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>.
</p><p>Recalling again that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =g^{-1}(X\beta )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>μ<!-- μ --></mi>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =g^{-1}(X\beta )}</annotation>
</semantics>
</math></span><img src="./ca0be1ae4c24a9d2b15ec6a737be9d2740c635dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.072ex; height:3.176ex;" alt="{\displaystyle \mu =g^{-1}(X\beta )}" loading="lazy"></span>, we derive the expressions for QL, QS and QI parametrized under <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>.
</p><p>Quasi-likelihood in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(\beta ,y)=\int _{y}^{\mu (\beta )}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q(\beta ,y)=\int _{y}^{\mu (\beta )}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}</annotation>
</semantics>
</math></span><img src="./4fc0f9ca2e8206b886e1cc0fbadd1b1567aec2c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.663ex; height:6.676ex;" alt="{\displaystyle Q(\beta ,y)=\int _{y}^{\mu (\beta )}{\frac {y-t}{\sigma ^{2}V(t)}}\,dt}" loading="lazy"></span></dd></dl>
<p>The QS as a function of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> is therefore
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{j}(\beta _{j})={\frac {\partial }{\partial \beta _{j}}}Q(\beta ,y)=\sum _{i=1}^{n}{\frac {\partial \mu _{i}}{\partial \beta _{j}}}{\frac {y_{i}-\mu _{i}(\beta _{j})}{\sigma ^{2}V(\mu _{i})}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U_{j}(\beta _{j})={\frac {\partial }{\partial \beta _{j}}}Q(\beta ,y)=\sum _{i=1}^{n}{\frac {\partial \mu _{i}}{\partial \beta _{j}}}{\frac {y_{i}-\mu _{i}(\beta _{j})}{\sigma ^{2}V(\mu _{i})}}}</annotation>
</semantics>
</math></span><img src="./021a5b06210f522753dc26711cf6525d9951a8f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.251ex; height:6.843ex;" alt="{\displaystyle U_{j}(\beta _{j})={\frac {\partial }{\partial \beta _{j}}}Q(\beta ,y)=\sum _{i=1}^{n}{\frac {\partial \mu _{i}}{\partial \beta _{j}}}{\frac {y_{i}-\mu _{i}(\beta _{j})}{\sigma ^{2}V(\mu _{i})}}}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(\beta )={\begin{bmatrix}U_{1}(\beta )\\U_{2}(\beta )\\\vdots \\\vdots \\U_{p}(\beta )\end{bmatrix}}=D^{T}V^{-1}{\frac {(y-\mu )}{\sigma ^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>U</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(\beta )={\begin{bmatrix}U_{1}(\beta )\\U_{2}(\beta )\\\vdots \\\vdots \\U_{p}(\beta )\end{bmatrix}}=D^{T}V^{-1}{\frac {(y-\mu )}{\sigma ^{2}}}}</annotation>
</semantics>
</math></span><img src="./86ecb106a10e4fae8fee0534c2abd95e39bd8fa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.468ex; margin-bottom: -0.203ex; width:36.367ex; height:18.509ex;" alt="{\displaystyle U(\beta )={\begin{bmatrix}U_{1}(\beta )\\U_{2}(\beta )\\\vdots \\\vdots \\U_{p}(\beta )\end{bmatrix}}=D^{T}V^{-1}{\frac {(y-\mu )}{\sigma ^{2}}}}" loading="lazy"></span></dd></dl>
<p>Where,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {D} _{n\times p}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{1}}{\partial \beta _{p}}}\\{\frac {\partial \mu _{2}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{2}}{\partial \beta _{p}}}\\\vdots \\\vdots \\{\frac {\partial \mu _{m}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{m}}{\partial \beta _{p}}}\end{bmatrix}}\underbrace {V} _{n\times n}=\operatorname {diag} (V(\mu _{1}),V(\mu _{2}),\ldots ,\ldots ,V(\mu _{n}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mi>D</mi>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>p</mi>
</mrow>
</munder>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mi>V</mi>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mrow>
</munder>
<mo>=</mo>
<mi>diag</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \underbrace {D} _{n\times p}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{1}}{\partial \beta _{p}}}\\{\frac {\partial \mu _{2}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{2}}{\partial \beta _{p}}}\\\vdots \\\vdots \\{\frac {\partial \mu _{m}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{m}}{\partial \beta _{p}}}\end{bmatrix}}\underbrace {V} _{n\times n}=\operatorname {diag} (V(\mu _{1}),V(\mu _{2}),\ldots ,\ldots ,V(\mu _{n}))}</annotation>
</semantics>
</math></span><img src="./5415d0e1607f2fcb2930156a4d298fb8476467b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:72.826ex; height:24.176ex;" alt="{\displaystyle \underbrace {D} _{n\times p}={\begin{bmatrix}{\frac {\partial \mu _{1}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{1}}{\partial \beta _{p}}}\\{\frac {\partial \mu _{2}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{2}}{\partial \beta _{p}}}\\\vdots \\\vdots \\{\frac {\partial \mu _{m}}{\partial \beta _{1}}}&amp;\cdots &amp;\cdots &amp;{\frac {\partial \mu _{m}}{\partial \beta _{p}}}\end{bmatrix}}\underbrace {V} _{n\times n}=\operatorname {diag} (V(\mu _{1}),V(\mu _{2}),\ldots ,\ldots ,V(\mu _{n}))}" loading="lazy"></span></dd></dl>
<p>The quasi-information matrix in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> is,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{b}=-{\frac {\partial U}{\partial \beta }}=\operatorname {Cov} (U(\beta ))={\frac {D^{T}V^{-1}D}{\sigma ^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>U</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>β<!-- β --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>Cov</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>β<!-- β --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msup>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mi>D</mi>
</mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{b}=-{\frac {\partial U}{\partial \beta }}=\operatorname {Cov} (U(\beta ))={\frac {D^{T}V^{-1}D}{\sigma ^{2}}}}</annotation>
</semantics>
</math></span><img src="./6766c547f0050999f2d9994eaad3fa147c4a9d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.905ex; height:6.176ex;" alt="{\displaystyle i_{b}=-{\frac {\partial U}{\partial \beta }}=\operatorname {Cov} (U(\beta ))={\frac {D^{T}V^{-1}D}{\sigma ^{2}}}}" loading="lazy"></span></dd></dl>
<p>Obtaining the score function and the information of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> allows for parameter estimation and inference in a similar manner as described in <a href="#Section_name">Application – weighted least squares</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Non-parametric_regression_analysis">Non-parametric regression analysis</h3></div>


<p>Non-parametric estimation of the variance function and its importance, has been discussed widely in the literature<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
In <a href="Non-parametric_regression" class="mw-redirect" title="Non-parametric regression">non-parametric regression</a> analysis, the goal is to express the expected value of your response variable(<b>y</b>) as a function of your predictors (<b>X</b>). That is we are looking to estimate a <b>mean</b> function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\operatorname {E} [y\mid X=x]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)=\operatorname {E} [y\mid X=x]}</annotation>
</semantics>
</math></span><img src="./e29ead89d2c10f6b08b90c2fcd7b9d5db085ebb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.731ex; height:2.843ex;" alt="{\displaystyle g(x)=\operatorname {E} [y\mid X=x]}" loading="lazy"></span> without assuming a parametric form. There are many forms of non-parametric <a href="Smoothing" title="Smoothing">smoothing</a> methods to help estimate the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>
</semantics>
</math></span><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy"></span>. An interesting approach is to also look at a non-parametric <b>variance function</b>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Var</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)}</annotation>
</semantics>
</math></span><img src="./fc43bd9217d51a89b3bf8e90203ef9a8d6ea7ff8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.121ex; height:2.843ex;" alt="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)}" loading="lazy"></span>. A non-parametric variance function allows one to look at the mean function as it relates to the variance function and notice patterns in the data.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Var</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}</annotation>
</semantics>
</math></span><img src="./c2acc3a048f2cb40914ad22955ddaf587a9cd6c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.222ex; height:3.343ex;" alt="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}" loading="lazy"></span></dd></dl>
<p>An example is detailed in the pictures to the right. The goal of the project was to determine (among other things) whether or not the predictor, <b>number of years in the major leagues</b> (baseball), had an effect on the response, <b>salary</b>, a player made. An initial scatter plot of the data indicates that there is heteroscedasticity in the data as the variance is not constant at each level of the predictor. Because we can visually detect the non-constant variance, it useful now to plot <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Var</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}</annotation>
</semantics>
</math></span><img src="./c2acc3a048f2cb40914ad22955ddaf587a9cd6c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.222ex; height:3.343ex;" alt="{\displaystyle g_{v}(x)=\operatorname {Var} (Y\mid X=x)=\operatorname {E} [y^{2}\mid X=x]-\left[\operatorname {E} [y\mid X=x]\right]^{2}}" loading="lazy"></span>, and look to see if the shape is indicative of any known distribution. One can estimate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [y^{2}\mid X=x]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [y^{2}\mid X=x]}</annotation>
</semantics>
</math></span><img src="./4bbf52f1d1ed16a90c812eb083566d7db520ee5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.437ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [y^{2}\mid X=x]}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\operatorname {E} [y\mid X=x]\right]^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mi mathvariant="normal">E</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">[</mo>
<mi>y</mi>
<mo>∣<!-- ∣ --></mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left[\operatorname {E} [y\mid X=x]\right]^{2}}</annotation>
</semantics>
</math></span><img src="./787ad25999fab0e8b63a4ad030838251e91632ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.725ex; height:3.343ex;" alt="{\displaystyle \left[\operatorname {E} [y\mid X=x]\right]^{2}}" loading="lazy"></span> using a general <a href="Smoothing" title="Smoothing">smoothing</a> method. The plot of the non-parametric smoothed variance function can give the researcher an idea of the relationship between the variance and the mean. The picture to the right indicates a quadratic relationship between the mean and the variance. As we saw above, the Gamma variance function is quadratic in the mean.
</p>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */


.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}


/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Muller1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Muller1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Muller1_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */


.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}


/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFMuller_and_Zhao1995" class="citation journal cs1">Muller and Zhao (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176324630">"On a semi parametric variance function model and a test for heteroscedasticity"</a>. <i>The Annals of Statistics</i>. <b>23</b> (3): <span class="nowrap">946–</span>967. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176324630">10.1214/aos/1176324630</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2242430">2242430</a>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMuller,_Stadtmuller_and_Yao2006" class="citation journal cs1">Muller, Stadtmuller and Yao (2006). "Functional Variance Processes". <i>Journal of the American Statistical Association</i>. <b>101</b> (475): <span class="nowrap">1007–</span>1018. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1198%2F016214506000000186">10.1198/016214506000000186</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27590778">27590778</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13712496">13712496</a>.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFWedderburn1974" class="citation journal cs1">Wedderburn, R.W.M. (1974). "Quasi-likelihood functions, generalized linear models, and the Gauss–Newton Method". <i>Biometrika</i>. <b>61</b> (3): <span class="nowrap">439–</span>447. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F61.3.439">10.1093/biomet/61.3.439</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2334725">2334725</a>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMcCullaghNelder,_John1989" class="citation book cs1">McCullagh, Peter; <a href="John_Nelder" title="John Nelder">Nelder, John</a> (1989). <i>Generalized Linear Models</i> (second&nbsp;ed.). London: Chapman and Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-412-31760-5</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)</span></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFMuller_and_StadtMuller1987" class="citation journal cs1">Muller and StadtMuller (1987). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350364">"Estimation of Heteroscedasticity in Regression Analysis"</a>. <i>The Annals of Statistics</i>. <b>15</b> (2): <span class="nowrap">610–</span>625. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350364">10.1214/aos/1176350364</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2241329">2241329</a>.</cite></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFCai_and_WangWang2008" class="citation journal cs1">Cai and Wang, T.; Wang, Lie (2008). "Adaptive Variance Function Estimation in Heteroscedastic Nonparametric Regression". <i>The Annals of Statistics</i>. <b>36</b> (5): <span class="nowrap">2025–</span>2054. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0810.4780">0810.4780</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008arXiv0810.4780C">2008arXiv0810.4780C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F07-AOS509">10.1214/07-AOS509</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2546470">2546470</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9184727">9184727</a>.</cite></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFRice_and_Silverman1991" class="citation journal cs1">Rice and Silverman (1991). "Estimating the Mean and Covariance structure nonparametrically when the data are curves". <i>Journal of the Royal Statistical Society</i>. <b>53</b> (1): <span class="nowrap">233–</span>243. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&nbsp;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2345738">2345738</a>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFMcCullaghNelder,_John1989" class="citation book cs1"><a href="Peter_McCullagh" title="Peter McCullagh">McCullagh, Peter</a>; <a href="John_Nelder" title="John Nelder">Nelder, John</a> (1989). <i>Generalized Linear Models</i> (second&nbsp;ed.). London: Chapman and Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-412-31760-5</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)</span></li>
<li><cite id="CITEREFHenrik_Madsen_and_Poul_Thyregod2011" class="citation book cs1">Henrik Madsen and Poul Thyregod (2011). <i>Introduction to General and Generalized Linear Models</i>. Chapman &amp; Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1-4200-9155-7</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="noviewer" typeof="mw:File"></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Variance_function" class="extiw external" title="commons:Category:Variance function">Variance function</a> at Wikimedia Commons</li></ul>
<div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1236075235">
/* start https://en.wikipedia.org/ */


.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}


/* end https://en.wikipedia.org/ */
</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>
<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>
</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mean" title="Mean">Mean</a>
<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>
<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>
<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>
<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>
<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>
<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>
<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>
<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>
<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>
<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>
<li><a href="Median" title="Median">Median</a></li>
<li><a href="Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>
<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>
<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>
<li><a href="Percentile" title="Percentile">Percentile</a></li>
<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>
<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>
<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>
<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>
<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>
<li><a href="L-moment" title="L-moment">L-moments</a></li>
<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>
<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>
<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>
<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>
<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>
<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>
<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>
<li><a href="Biplot" title="Biplot">Biplot</a></li>
<li><a href="Box_plot" title="Box plot">Box plot</a></li>
<li><a href="Control_chart" title="Control chart">Control chart</a></li>
<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>
<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>
<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>
<li><a href="Histogram" title="Histogram">Histogram</a></li>
<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>
<li><a href="Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li>
<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>
<li><a href="Run_chart" title="Run chart">Run chart</a></li>
<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>
<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>
<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>
<li><a href="Missing_data" title="Missing data">Missing data</a></li>
<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>
<li><a href="Statistical_population" title="Statistical population">Population</a></li>
<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>
<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>
<li><a href="Statistic" title="Statistic">Statistic</a></li>
<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>
<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>
<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>
<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>
<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>
<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>
<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>
<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>
<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>
<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>
<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>
<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>
<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>
<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>
<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>
<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>
<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>
<li><a href="Statistic" title="Statistic">Statistic</a></li>
<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>
<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>
<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>
<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>
<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>
<li><a href="Statistical_model" title="Statistical model">Statistical model</a>
<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>
<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>
<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>
<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>
<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>
<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>
<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>
<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a>&nbsp;<a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>
<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>
<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>
<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>
<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>
<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>
<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="U-statistic" title="U-statistic">U</a></li>
<li><a href="V-statistic" title="V-statistic">V</a></li></ul></li>
<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>
<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>
<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>
<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>
<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>
<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>
<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>
<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>
<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>
<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>
<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>
<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>
<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>
<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>
<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>
<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>
<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>
<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>
<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>
<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>
<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>
<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>
<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- &amp; 2-tails</a></li>
<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>
<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>
<li><a href="Permutation_test" title="Permutation test">Permutation test</a>
<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>
<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>
<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>
<li><a href="Wald_test" title="Wald test">Wald</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>
<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>
<li><a href="F-test" title="F-test"><i>F</i>-test</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>
<li><a href="G-test" title="G-test"><i>G</i>-test</a></li>
<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>
<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>
<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>
<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>
<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>
<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>
<li><a href="Model_selection" title="Model selection">Model selection</a>
<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>
<li><a href="Akaike_information_criterion" title="Akaike information criterion">AIC</a></li>
<li><a href="Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Sign_test" title="Sign test">Sign</a>
<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>
<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>
<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>
<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>
<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>
<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>
<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>
<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>
<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>
<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>
<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>
<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>
<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>
<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>
<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>
<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="Confounding" title="Confounding">Confounding variable</a></li>
<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>
<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>
<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>
<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>
<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>
<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>
<li><a href="General_linear_model" title="General linear model">General linear model</a></li>
<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>
<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>
<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>
<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>
<li><a href="Robust_regression" title="Robust regression">Robust</a></li>
<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>
<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a>&nbsp;/ <a href="Binomial_regression" title="Binomial regression">Binomial</a>&nbsp;/ <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>
<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>
<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>
<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a>&nbsp;/ <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a>&nbsp;/ <a href="Time_series" title="Time series">time-series</a>&nbsp;/ <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>
<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>
<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>
<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>
<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>
<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>
<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>
<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>
<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>
<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>
<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>
<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>
<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>
<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>
<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>
<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>
<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>
<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>
<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>
<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>
<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>
<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>
<li><a href="Structural_break" title="Structural break">Structural break</a></li>
<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>
<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>
<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>
<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>
<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>
<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>
<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>
<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>
<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>
<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>
<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>
<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>
<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>
<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>
<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>
<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>
<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>
<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>
<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a>&nbsp;/ <a href="Clinical_study_design" title="Clinical study design">studies</a></li>
<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>
<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>
<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>
<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>
<li><a href="Statistical_process_control" title="Statistical process control">Process</a>&nbsp;/ <a href="Quality_control" title="Quality control">quality control</a></li>
<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>
<li><a href="System_identification" title="System identification">System identification</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>
<li><a href="Census" title="Census">Census</a></li>
<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>
<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>
<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>
<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>
<li><a href="National_accounts" title="National accounts">National accounts</a></li>
<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>
<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>
<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>
<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>
<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>
<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>
<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>
<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>
<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>
<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>
</div></td></tr></tbody></table></div></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2023-09-14" href="https://en.wikipedia.org/wiki/?title=Variance_function&amp;oldid=1175406517">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>

</body></html>