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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Loss function</span></span>

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<p>In <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a> and <a href="Decision_theory" title="Decision theory">decision theory</a>, a <b>loss function</b> or <b>cost function</b> (sometimes also called an error function)<sup id="cite_ref-ttf2001_1-0" class="reference"><a href="#cite_note-ttf2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a function that maps an <a href="Event_(probability_theory)" title="Event (probability theory)">event</a> or values of one or more variables onto a <a href="Real_number" title="Real number">real number</a> intuitively representing some "cost" associated with the event. An <a href="Optimization_problem" title="Optimization problem">optimization problem</a> seeks to minimize a loss function. An <b>objective function</b> is either a loss function or its opposite (in specific domains, variously called a <a href="Reward_function" class="mw-redirect" title="Reward function">reward function</a>, a <a href="Profit_function" class="mw-redirect" title="Profit function">profit function</a>, a <a href="Utility_function" class="mw-redirect" title="Utility function">utility function</a>, a <a href="Fitness_function" title="Fitness function">fitness function</a>, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy.

</p><p>In statistics, typically a loss function is used for <a href="Parameter_estimation" class="mw-redirect" title="Parameter estimation">parameter estimation</a>, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a>, was reintroduced in statistics by <a href="Abraham_Wald" title="Abraham Wald">Abraham Wald</a> in the middle of the 20th century.<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 the context of <a href="Economics" title="Economics">economics</a>, for example, this is usually <a href="Economic_cost" title="Economic cost">economic cost</a> or <a href="Regret_(decision_theory)" title="Regret (decision theory)">regret</a>. In <a href="Statistical_classification" title="Statistical classification">classification</a>, it is the penalty for an incorrect classification of an example. In <a href="Actuarial_science" title="Actuarial science">actuarial science</a>, it is used in an insurance context to model benefits paid over premiums, particularly since the works of <a href="Harald_Cram%C3%A9r" title="Harald Cramér">Harald Cramér</a> in the 1920s.<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> In <a href="Optimal_control" title="Optimal control">optimal control</a>, the loss is the penalty for failing to achieve a desired value. In <a href="Financial_risk_management" title="Financial risk management">financial risk management</a>, the function is mapped to a monetary loss.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Regret">Regret</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Regret_(decision_theory)" title="Regret (decision theory)">Regret (decision theory)</a></div>

<p><a href="Leonard_J._Savage" class="mw-redirect" title="Leonard J. Savage">Leonard J. Savage</a> argued that using non-Bayesian methods such as <a href="Minimax" title="Minimax">minimax</a>, the loss function should be based on the idea of <i><a href="Regret_(decision_theory)" title="Regret (decision theory)">regret</a></i>, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known.

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<div class="mw-heading mw-heading3"><h3 id="Quadratic_loss_function">Quadratic loss function</h3></div>

<p>The use of a <a href="Quadratic_function" title="Quadratic function">quadratic</a> loss function is common, for example when using <a href="Least_squares" title="Least squares">least squares</a> techniques. It is often more mathematically tractable than other loss functions because of the properties of <a href="Variance" title="Variance">variances</a>, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is <i>t</i>, then a quadratic loss function is

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<annotation encoding="application/x-tex">{\displaystyle \lambda (x)=C(t-x)^{2}\;}</annotation>

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</math></span><img src="./cf4beff1dc104f16784ac54e594efbdaa72480b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.877ex; height:3.176ex;" alt="{\displaystyle \lambda (x)=C(t-x)^{2}\;}" loading="lazy"></span></dd></dl>

<p>for some constant <i>C</i>; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the <b>squared error loss</b> (<b>SEL</b>).<sup id="cite_ref-ttf2001_1-1" class="reference"><a href="#cite_note-ttf2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Many common <a href="Statistic" title="Statistic">statistics</a>, including <a href="T-test" class="mw-redirect" title="T-test">t-tests</a>, <a href="Regression_analysis" title="Regression analysis">regression</a> models, <a href="Design_of_experiments" title="Design of experiments">design of experiments</a>, and much else, use <a href="Least_squares" title="Least squares">least squares</a> methods applied using <a href="Linear_regression" title="Linear regression">linear regression</a> theory, which is based on the quadratic loss function.

</p><p>The quadratic loss function is also used in <a href="Linear-quadratic_regulator" class="mw-redirect" title="Linear-quadratic regulator">linear-quadratic optimal control problems</a>. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a <a href="Quadratic_form" title="Quadratic form">quadratic form</a> in the deviations of the variables of interest from their desired values; this approach is <a href="Closed-form_expression" title="Closed-form expression">tractable</a> because it results in linear <a href="First-order_condition" class="mw-redirect" title="First-order condition">first-order conditions</a>. In the context of <a href="Stochastic_control" title="Stochastic control">stochastic control</a>, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the <a href="Huber_loss" title="Huber loss">Huber</a>, log-cosh and SMAE losses are used when the data has many large outliers.

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<div class="mw-heading mw-heading3"><h3 id="0-1_loss_function">0-1 loss function</h3></div>

<p>In <a href="Statistics" title="Statistics">statistics</a> and <a href="Decision_theory" title="Decision theory">decision theory</a>, a frequently used loss function is the <i>0-1 loss function</i>

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<annotation encoding="application/x-tex">{\displaystyle L({\hat {y}},y)=\left[{\hat {y}}\neq y\right]}</annotation>

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</math></span><img src="./9799c7b708b5760abff4fd57536d2cc5ddf8cf04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.832ex; height:2.843ex;" alt="{\displaystyle L({\hat {y}},y)=\left[{\hat {y}}\neq y\right]}" loading="lazy"></span></dd></dl>

<p>using <a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a> notation, i.e. it evaluates to 1 when <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 {y}}\neq y}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}\neq y}</annotation>

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</math></span><img src="./fbb724d211b93790aba8529caa8ac1f2293d789e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.556ex; height:2.676ex;" alt="{\displaystyle {\hat {y}}\neq y}" loading="lazy"></span>, and 0 otherwise.

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<div class="mw-heading mw-heading2"><h2 id="Constructing_loss_and_objective_functions">Constructing loss and objective functions</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Scoring_rule" title="Scoring rule">Scoring rule</a></div>

<p>In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also <a href="Utility" title="Utility">utility</a> function) in a form suitable for optimization — the problem that <a href="Ragnar_Frisch" title="Ragnar Frisch">Ragnar Frisch</a> has highlighted in his <a href="Nobel_Prize" title="Nobel Prize">Nobel Prize</a> lecture.<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>

The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences.<sup id="cite_ref-TangianGruber1997_5-0" class="reference"><a href="#cite_note-TangianGruber1997-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-TangianGruber2002_6-0" class="reference"><a href="#cite_note-TangianGruber2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

In particular, <a href="Andranik_Tangian" title="Andranik Tangian">Andranik Tangian</a> showed that the most usable objective functions — quadratic and additive — are determined by a few <a href="Principle_of_indifference" title="Principle of indifference">indifference</a> points. He used this property in the models for constructing these objective functions from either <a href="Ordinal_utility" title="Ordinal utility">ordinal</a> or <a href="Cardinal_utility" title="Cardinal utility">cardinal</a> data that were elicited through computer-assisted interviews with decision makers.<sup id="cite_ref-Tangian2002_7-0" class="reference"><a href="#cite_note-Tangian2002-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tangian2004additiveUtility_8-0" class="reference"><a href="#cite_note-Tangian2004additiveUtility-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities<sup id="cite_ref-Tangian2004universityBudgets_9-0" class="reference"><a href="#cite_note-Tangian2004universityBudgets-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

and the European subsidies for equalizing unemployment rates among 271 German regions.<sup id="cite_ref-Tangian2008RegionalEnemployment_10-0" class="reference"><a href="#cite_note-Tangian2008RegionalEnemployment-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Expected_loss">Expected loss</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></div>

<p>In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable <i>X</i>.

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<div class="mw-heading mw-heading3"><h3 id="Statistics">Statistics</h3></div>

<p>Both <a href="Frequentist_probability" title="Frequentist probability">frequentist</a> and <a href="Bayesian_probability" title="Bayesian probability">Bayesian</a> statistical theory involve making a decision based on the <a href="Expected_value" title="Expected value">expected value</a> of the loss function; however, this quantity is defined differently under the two paradigms.

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<div class="mw-heading mw-heading4"><h4 id="Frequentist_expected_loss">Frequentist expected loss</h4></div>

<p>We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, <i>P</i><sub><i>θ</i></sub>, of the observed data, <i>X</i>. This is also referred to as the <b>risk function</b><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> of the decision rule <i>δ</i> and the parameter <i>θ</i>. Here the decision rule depends on the outcome of <i>X</i>. The risk function is given by:

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<annotation encoding="application/x-tex">{\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).}</annotation>

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</math></span><img src="./490551852e01dfc66a2a0dd7a55078135d00c554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.889ex; height:5.676ex;" alt="{\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).}" loading="lazy"></span></dd></dl>

<p>Here, <i>θ</i> is a fixed but possibly unknown state of nature, <i>X</i> is a vector of observations stochastically drawn from a <a href="Statistical_population" title="Statistical population">population</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 \operatorname {E} _{\theta }}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }}</annotation>

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</math></span><img src="./9956e06f5dac5bc226e3dc8faba2290036d82081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.586ex; height:2.509ex;" alt="{\displaystyle \operatorname {E} _{\theta }}" loading="lazy"></span> is the expectation over all population values of <i>X</i>, <i>dP</i><sub><i>θ</i></sub> is a <a href="Probability_measure" title="Probability measure">probability measure</a> over the event space of <i>X</i> (parametrized by <i>θ</i>) and the integral is evaluated over the entire <a href="Support_(measure_theory)" title="Support (measure theory)">support</a> of <i>X</i>.

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<div class="mw-heading mw-heading4"><h4 id="Bayes_Risk">Bayes Risk</h4></div>

<p>In a Bayesian approach, the expectation is calculated using the <a href="Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a> <span class="texhtml mvar" style="font-style:italic;">π</span><sup>*</sup> of the parameter <i>θ</i>:

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<annotation encoding="application/x-tex">{\displaystyle \rho (\pi ^{*},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{*}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{*}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})}</annotation>

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</math></span><img src="./60125739cdefb1e35d50fdb7f3223c1beefe1536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:78.572ex; height:5.676ex;" alt="{\displaystyle \rho (\pi ^{*},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{*}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{*}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})}" loading="lazy"></span></dd></dl>

<p>where m(x) is known as the <i>predictive likelihood</i> wherein θ has been "integrated out," <span class="texhtml mvar" style="font-style:italic;">π</span><sup>*</sup> (θ | x) is the posterior distribution, and the order of integration has been changed. One then should choose the action <i>a<sup>*</sup></i> which minimises this expected loss, which is referred to as <i>Bayes Risk</i>.

In the latter equation, the integrand inside dx is known as the <i>Posterior Risk</i>, and minimising it with respect to decision <i>a</i> also minimizes the overall Bayes Risk. This optimal decision, <i>a<sup>*</sup></i> is known as the <i>Bayes (decision) Rule</i> - it minimises the average loss over all possible states of nature θ, over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ.

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<div class="mw-heading mw-heading4"><h4 id="Examples_in_statistics">Examples in statistics</h4></div>

<ul><li>For a scalar parameter <i>θ</i>, a decision function whose output <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 {\theta }}}">

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</math></span><img src="./f0eaae56d74c5844e86caeed8ae205ff9e413bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}" loading="lazy"></span> is an estimate of <i>θ</i>, and a quadratic loss function (<a href="Squared_error_loss" class="mw-redirect" title="Squared error loss">squared error loss</a>) <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},}">

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</math></span></span> the risk function becomes the <a href="Mean_squared_error" title="Mean squared error">mean squared error</a> of the estimate, <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\theta ,{\hat {\theta }})=\operatorname {E} _{\theta }\left[(\theta -{\hat {\theta }})^{2}\right].}">

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<annotation encoding="application/x-tex">{\displaystyle R(\theta ,{\hat {\theta }})=\operatorname {E} _{\theta }\left[(\theta -{\hat {\theta }})^{2}\right].}</annotation>

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</math></span></span>An <a href="Estimator" title="Estimator">Estimator</a> found by minimizing the <a href="Mean_squared_error" title="Mean squared error">Mean squared error</a> estimates the <a href="Posterior_distribution" class="mw-redirect" title="Posterior distribution">Posterior distribution</a>'s mean.</li>

<li>In <a href="Density_estimation" title="Density estimation">density estimation</a>, the unknown parameter is <a href="Probability_density_function" title="Probability density function">probability density</a> itself. The loss function is typically chosen to be a <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> in an appropriate <a href="Function_space" title="Function space">function space</a>. For example, for <a href="L2_norm" class="mw-redirect" title="L2 norm"><i>L</i><sup>2</sup> norm</a>, <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(f,{\hat {f}})=\|f-{\hat {f}}\|_{2}^{2}\,,}">

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<annotation encoding="application/x-tex">{\displaystyle L(f,{\hat {f}})=\|f-{\hat {f}}\|_{2}^{2}\,,}</annotation>

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</math></span></span> the risk function becomes the <a href="Mean_integrated_squared_error" title="Mean integrated squared error">mean integrated squared error</a> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(f,{\hat {f}})=\operatorname {E} \left(\|f-{\hat {f}}\|^{2}\right).\,}">

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<annotation encoding="application/x-tex">{\displaystyle R(f,{\hat {f}})=\operatorname {E} \left(\|f-{\hat {f}}\|^{2}\right).\,}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Economic_choice_under_uncertainty">Economic choice under uncertainty</h3></div>

<p>In economics, decision-making under uncertainty is often modelled using the <a href="Von_Neumann%E2%80%93Morgenstern_utility_function" class="mw-redirect" title="Von Neumann–Morgenstern utility function">von Neumann–Morgenstern utility function</a> of the uncertain variable of interest, such as end-of-period wealth. Since the value of this variable is uncertain, so is the value of the utility function; it is the expected value of utility that is maximized.

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<div class="mw-heading mw-heading2"><h2 id="Decision_rules">Decision rules</h2></div>

<p>A <a href="Decision_rule" title="Decision rule">decision rule</a> makes a choice using an optimality criterion. Some commonly used criteria are:

</p>

<ul><li><b><a href="Minimax" title="Minimax">Minimax</a></b>: Choose the decision rule with the lowest worst loss — that is, minimize the worst-case (maximum possible) loss: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {\delta }{\operatorname {arg\,min} }}\ \max _{\theta \in \Theta }\ R(\theta ,\delta ).}">

</mrow>

</munder>

</mrow>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underset {\delta }{\operatorname {arg\,min} }}\ \max _{\theta \in \Theta }\ R(\theta ,\delta ).}</annotation>

</semantics>

</math></span></span></li>

<li><b><a href="Invariant_estimator" title="Invariant estimator">Invariance</a></b>: Choose the decision rule which satisfies an invariance requirement.</li>

<li>Choose the decision rule with the lowest average loss (i.e. minimize the <a href="Expected_value" title="Expected value">expected value</a> of the loss function): <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {\delta }{\operatorname {arg\,min} }}\operatorname {E} _{\theta \in \Theta }[R(\theta ,\delta )]={\underset {\delta }{\operatorname {arg\,min} }}\ \int _{\theta \in \Theta }R(\theta ,\delta )\,p(\theta )\,d\theta .}">

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underset {\delta }{\operatorname {arg\,min} }}\operatorname {E} _{\theta \in \Theta }[R(\theta ,\delta )]={\underset {\delta }{\operatorname {arg\,min} }}\ \int _{\theta \in \Theta }R(\theta ,\delta )\,p(\theta )\,d\theta .}</annotation>

</semantics>

</math></span></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Selecting_a_loss_function">Selecting a loss function</h2></div>

<p>Sound statistical practice requires selecting an estimator consistent with the actual acceptable variation experienced in the context of a particular applied problem. Thus, in the applied use of loss functions, selecting which statistical method to use to model an applied problem depends on knowing the losses that will be experienced from being wrong under the problem's particular circumstances.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>A common example involves estimating "<a href="Location_parameter" title="Location parameter">location</a>". Under typical statistical assumptions, the <a href="Mean" title="Mean">mean</a> or average is the statistic for estimating location that minimizes the expected loss experienced under the <a href="Least_squares" title="Least squares">squared-error</a> loss function, while the <a href="Median" title="Median">median</a> is the estimator that minimizes expected loss experienced under the absolute-difference loss function. Still different estimators would be optimal under other, less common circumstances.

</p><p>In economics, when an agent is <a href="Risk_neutral" class="mw-redirect" title="Risk neutral">risk neutral</a>, the objective function is simply expressed as the expected value of a monetary quantity, such as profit, income, or end-of-period wealth. For <a href="Risk_aversion" title="Risk aversion">risk-averse</a> or <a href="Risk-loving" class="mw-redirect" title="Risk-loving">risk-loving</a> agents, loss is measured as the negative of a <a href="Utility" title="Utility">utility function</a>, and the objective function to be optimized is the expected value of utility.

</p><p>Other measures of cost are possible, for example <a href="Mortality_rate" title="Mortality rate">mortality</a> or <a href="Morbidity" class="mw-redirect" title="Morbidity">morbidity</a> in the field of <a href="Public_health" title="Public health">public health</a> or <a href="Safety_engineering" title="Safety engineering">safety engineering</a>.

</p><p>For most <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">optimization algorithms</a>, it is desirable to have a loss function that is globally <a href="Continuous_function" title="Continuous function">continuous</a> and <a href="Differentiable_function" title="Differentiable function">differentiable</a>.

</p><p>Two very commonly used loss functions are the <a href="Mean_squared_error" title="Mean squared error">squared loss</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 L(a)=a^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(a)=a^{2}}</annotation>

</semantics>

</math></span><img src="./63618aaa8422af5bf2aa2f5051f2ff305d9af5b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.004ex; height:3.176ex;" alt="{\displaystyle L(a)=a^{2}}" loading="lazy"></span>, and the <a href="Absolute_deviation" class="mw-redirect" title="Absolute deviation">absolute loss</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 L(a)=|a|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(a)=|a|}</annotation>

</semantics>

</math></span><img src="./a49ef4dd7fa7d5804d7aca226f471e2198cab648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle L(a)=|a|}" loading="lazy"></span>. However the absolute loss has the disadvantage that it is not differentiable at <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 a=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=0}</annotation>

</semantics>

</math></span><img src="./90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy"></span>. The squared loss has the disadvantage that it has the tendency to be dominated by <a href="Outlier" title="Outlier">outliers</a>—when summing over a set 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 a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>'s (as 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="{\textstyle \sum _{i=1}^{n}L(a_{i})}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}L(a_{i})}</annotation>

</semantics>

</math></span><img src="./9270305d9660c39f850d93ec13bff570f8903bd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.163ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}L(a_{i})}" loading="lazy"></span>), the final sum tends to be the result of a few particularly large <i>a</i>-values, rather than an expression of the average <i>a</i>-value.

</p><p>The choice of a loss function is not arbitrary. It is very restrictive and sometimes the loss function may be characterized by its desirable properties.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Among the choice principles are, for example, the requirement of completeness of the class of symmetric statistics in the case of <a href="I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> observations, the principle of complete information, and some others.

</p><p><a href="W._Edwards_Deming" title="W. Edwards Deming">W. Edwards Deming</a> and <a href="Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Nicholas Taleb</a> argue that empirical reality, not nice mathematical properties, should be the sole basis for selecting loss functions, and real losses often are not mathematically nice and are not differentiable, continuous, symmetric, etc. For example, a person who arrives before a plane gate closure can still make the plane, but a person who arrives after can not, a discontinuity and asymmetry which makes arriving slightly late much more costly than arriving slightly early. In drug dosing, the cost of too little drug may be lack of efficacy, while the cost of too much may be tolerable toxicity, another example of asymmetry. Traffic, pipes, beams, ecologies, climates, etc. may tolerate increased load or stress with little noticeable change up to a point, then become backed up or break catastrophically. These situations, Deming and Taleb argue, are common in real-life problems, perhaps more common than classical smooth, continuous, symmetric, differentials cases.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Bayesian_regret" title="Bayesian regret">Bayesian regret</a></li>

<li><a href="Loss_functions_for_classification" title="Loss functions for classification">Loss functions for classification</a></li>

<li><a href="Discounted_maximum_loss" title="Discounted maximum loss">Discounted maximum loss</a></li>

<li><a href="Hinge_loss" title="Hinge loss">Hinge loss</a></li>

<li><a href="Scoring_rule" title="Scoring rule">Scoring rule</a></li>

<li><a href="Statistical_risk" title="Statistical risk">Statistical risk</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFPfanzagl1994" class="citation book cs1">Pfanzagl, J. (1994). <i>Parametric Statistical Theory</i>. Berlin: Walter de Gruyter. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-013863-4</bdi>.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Detailed information on mathematical principles of the loss function choice is given in Chapter 2 of the book <cite id="CITEREFKlebanovRachevFabozzi2009" class="citation book cs1">Klebanov, B.; Rachev, Svetlozat T.; Fabozzi, Frank J. (2009). <i>Robust and Non-Robust Models in Statistics</i>. New York: Nova Scientific Publishers, Inc.</cite> (and references there).</span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFDeming2000" class="citation book cs1">Deming, W. Edwards (2000). <i>Out of the Crisis</i>. The MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262541152</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFAretzBartram,_Söhnke_M.Pope,_Peter_F.2011" class="citation journal cs1">Aretz, Kevin; Bartram, Söhnke M.; Pope, Peter F. (April–June 2011). <a rel="nofollow" class="external text" href="https://mpra.ub.uni-muenchen.de/47343/1/MPRA_paper_47343.pdf">"Asymmetric Loss Functions and the Rationality of Expected Stock Returns"</a> <span class="cs1-format">(PDF)</span>. <i>International Journal of Forecasting</i>. <b>27</b> (2): <span class="nowrap">413–</span>437. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijforecast.2009.10.008">10.1016/j.ijforecast.2009.10.008</a>. <a href="SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=889323">889323</a>.</cite></li>

<li><cite id="CITEREFBerger1985" class="citation book cs1"><a href="James_Berger_(statistician)" class="mw-redirect" title="James Berger (statistician)">Berger, James O.</a> (1985). <i>Statistical decision theory and Bayesian Analysis</i> (2nd ed.). New York: Springer-Verlag. <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/1985sdtb.book.....B">1985sdtb.book.....B</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96098-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0804611">0804611</a>.</cite></li></ul>

<ul><li><cite id="CITEREFCecchetti2000" class="citation journal cs1">Cecchetti, S. (2000). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/5216117">"Making monetary policy: Objectives and rules"</a>. <i>Oxford Review of Economic Policy</i>. <b>16</b> (4): <span class="nowrap">43–</span>59. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Foxrep%2F16.4.43">10.1093/oxrep/16.4.43</a>.</cite></li></ul>

<ul><li><cite id="CITEREFHorowitz1987" class="citation journal cs1">Horowitz, Ann R. (1987). "Loss functions and public policy". <i>Journal of Macroeconomics</i>. <b>9</b> (4): <span class="nowrap">489–</span>504. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0164-0704%2887%2990016-4">10.1016/0164-0704(87)90016-4</a>.</cite></li></ul>

<ul><li><cite id="CITEREFWaud1976" class="citation journal cs1">Waud, Roger N. (1976). "Asymmetric Policymaker Utility Functions and Optimal Policy under Uncertainty". <i>Econometrica</i>. <b>44</b> (1): <span class="nowrap">53–</span>66. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1911380">10.2307/1911380</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1911380">1911380</a>.</cite></li></ul>

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</style><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="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>

</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="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="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 uncollapsed 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> <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="Optimal_decision" title="Optimal decision">Optimal decision</a>

<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>

</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- & 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>

</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>

</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="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>

</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="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 mw-collapsed 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> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <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> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <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">

</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> / <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> / <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>

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<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Differentiable_computing254" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Differentiable_computing254" style="font-size:114%;margin:0 4em">Differentiable computing</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differentiable_function" title="Differentiable function">General</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><b><a href="Differentiable_programming" title="Differentiable programming">Differentiable programming</a></b></li>

<li><a href="Information_geometry" title="Information geometry">Information geometry</a></li>

<li><a href="Statistical_manifold" title="Statistical manifold">Statistical manifold</a></li>

<li><a href="Automatic_differentiation" title="Automatic differentiation">Automatic differentiation</a></li>

<li><a href="Neuromorphic_computing" title="Neuromorphic computing">Neuromorphic computing</a></li>

<li><a href="Pattern_recognition" title="Pattern recognition">Pattern recognition</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li>

<li><a href="Inductive_bias" title="Inductive bias">Inductive bias</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hardware</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Memristor" title="Memristor">Memristor</a></li>

<li><a href="SpiNNaker" title="SpiNNaker">SpiNNaker</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software libraries</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="TensorFlow" title="TensorFlow">TensorFlow</a></li>

<li><a href="PyTorch" title="PyTorch">PyTorch</a></li>

<li><a href="Keras" title="Keras">Keras</a></li>

<li><a href="Scikit-learn" title="Scikit-learn">scikit-learn</a></li>

<li><a href="Theano_(software)" title="Theano (software)">Theano</a></li>

<li><a href="MindSpore" title="MindSpore">MindSpore</a></li></ul>

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<ul><li><span class="noviewer" typeof="mw:File"></span> Portals

<ul><li><a href="Portal%3AComputer_programming" title="Portal:Computer programming">Computer programming</a></li>

<li><a href="Portal%3ATechnology" title="Portal:Technology">Technology</a></li></ul></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-07-25" href="https://en.wikipedia.org/wiki/?title=Loss_function&oldid=1302457864">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.

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