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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Efficiency (statistics)</span></span>

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<p>In statistics, <b>efficiency</b> is a measure of quality of an <a href="Estimator" title="Estimator">estimator</a>, of an experimental design,<sup id="cite_ref-FOOTNOTEEveritt2002128_1-0" class="reference"><a href="#cite_note-FOOTNOTEEveritt2002128-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> or of a <a href="Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a> procedure.<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> Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achieve the <a href="Cram%C3%A9r%E2%80%93Rao_bound" title="Cramér–Rao bound">Cramér–Rao bound</a>.

An <i>efficient estimator</i> is characterized by having the smallest possible <a href="Variance" title="Variance">variance</a>, indicating that there is a small <a href="Deviance_(statistics)" title="Deviance (statistics)">deviance</a> between the estimated value and the "true" value in the <a href="L2_norm" class="mw-redirect" title="L2 norm">L2 norm</a> sense.<sup id="cite_ref-FOOTNOTEEveritt2002128_1-1" class="reference"><a href="#cite_note-FOOTNOTEEveritt2002128-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The <b>relative efficiency</b> of two procedures is the ratio of their efficiencies, although often this concept is used where the comparison is made between a given procedure and a notional "best possible" procedure. The efficiencies and the relative efficiency of two procedures theoretically depend on the sample size available for the given procedure, but it is often possible to use the <b>asymptotic relative efficiency</b> (defined as the limit of the relative efficiencies as the sample size grows) as the principal comparison measure.

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

<p>The efficiency of an <a href="Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> <a href="Estimator" title="Estimator">estimator</a>, <i>T</i>, of a <a href="Statistical_parameter" title="Statistical parameter">parameter</a> <i>θ</i> is defined as <sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle e(T)={\frac {1/{\mathcal {I}}(\theta )}{\operatorname {var} (T)}}}</annotation>

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</math></span><img src="./181f1a5ead95b2c549be2d1d2f09583f5cb9b967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.211ex; height:6.509ex;" alt="{\displaystyle e(T)={\frac {1/{\mathcal {I}}(\theta )}{\operatorname {var} (T)}}}" 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 {\mathcal {I}}(\theta )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}(\theta )}</annotation>

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</math></span><img src="./93d0e554bc0fb296dac5ded2a7be914f4398543e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.069ex; width:4.461ex; height:2.843ex;" alt="{\displaystyle {\mathcal {I}}(\theta )}" loading="lazy"></span> is the <a href="Fisher_information" title="Fisher information">Fisher information</a> of the sample. Thus <i>e</i>(<i>T</i>) is the minimum possible variance for an unbiased estimator divided by its actual variance. The <a href="Cram%C3%A9r%E2%80%93Rao_bound" title="Cramér–Rao bound">Cramér–Rao bound</a> can be used to prove that <i>e</i>(<i>T</i>) ≤ 1.

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<div class="mw-heading mw-heading3"><h3 id="Efficient_estimators">Efficient estimators</h3></div>

<p>An <b>efficient estimator</b> is an <a href="Estimator" title="Estimator">estimator</a> that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular <a href="Loss_function" title="Loss function">loss function</a> — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. The most common choice of the loss function is <a href="Quadratic_loss_function" class="mw-redirect" title="Quadratic loss function">quadratic</a>, resulting in the <a href="Mean_squared_error" title="Mean squared error">mean squared error</a> criterion of optimality.<sup id="cite_ref-FOOTNOTEEveritt2002[httpsarchiveorgdetailscambridgediction00everpagen135_128]_4-0" class="reference"><a href="#cite_note-FOOTNOTEEveritt2002[httpsarchiveorgdetailscambridgediction00everpagen135_128]-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>In general, the spread of an estimator around the parameter θ is a measure of estimator efficiency and performance. This performance can be calculated by finding the mean squared error. More formally, let <i>T</i> be an estimator for the parameter <i>θ</i>. The mean squared error of <i>T</i> is the value <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 {MSE} (T)=E[(T-\theta )^{2}]}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} (T)=E[(T-\theta )^{2}]}</annotation>

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</math></span><img src="./ba0fa50239427e73a435244a58b28e3596099390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.05ex; height:3.176ex;" alt="{\displaystyle \operatorname {MSE} (T)=E[(T-\theta )^{2}]}" loading="lazy"></span>, which can be decomposed as a sum of its variance and bias:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {MSE} (T)&=\operatorname {E} [(T-\theta )^{2}]=\operatorname {E} [(T-\operatorname {E} [T]+\operatorname {E} [T]-\theta )^{2}]\\[5pt]&=\operatorname {E} [(T-\operatorname {E} [T])^{2}]+2E[T-E[T]](\operatorname {E} [T]-\theta )+(\operatorname {E} [T]-\theta )^{2}\\[5pt]&=\operatorname {var} (T)+(\operatorname {E} [T]-\theta )^{2}\end{aligned}}}</annotation>

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</math></span><img src="./df45d4c039ab576a6b5d819378eb7add75dc72e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:67.686ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {MSE} (T)&=\operatorname {E} [(T-\theta )^{2}]=\operatorname {E} [(T-\operatorname {E} [T]+\operatorname {E} [T]-\theta )^{2}]\\[5pt]&=\operatorname {E} [(T-\operatorname {E} [T])^{2}]+2E[T-E[T]](\operatorname {E} [T]-\theta )+(\operatorname {E} [T]-\theta )^{2}\\[5pt]&=\operatorname {var} (T)+(\operatorname {E} [T]-\theta )^{2}\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>An estimator <i>T</i><sub>1</sub> performs better than an estimator <i>T</i><sub>2</sub> if <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 {MSE} (T_{1})<\operatorname {MSE} (T_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} (T_{1})<\operatorname {MSE} (T_{2})}</annotation>

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</math></span><img src="./8879cdb08cfc49c32a4110eee6182cd6157afc2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.553ex; height:2.843ex;" alt="{\displaystyle \operatorname {MSE} (T_{1})<\operatorname {MSE} (T_{2})}" loading="lazy"></span>.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For a more specific case, if <i>T</i><sub>1</sub> and <i>T<sub>2</sub> </i>are two unbiased estimators for the same parameter θ, then the variance can be compared to determine performance. In this case, <i>T</i><sub>2</sub> is <i>more efficient</i> than <i>T</i><sub>1</sub> if the variance of <i>T</i><sub>2</sub> is <i>smaller</i> than the variance of <i>T</i><sub>1</sub>, i.e. <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} (T_{1})>\operatorname {var} (T_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {var} (T_{1})>\operatorname {var} (T_{2})}</annotation>

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</math></span><img src="./2826d92002db92bd061e627be39675580c7748a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.144ex; height:2.843ex;" alt="{\displaystyle \operatorname {var} (T_{1})>\operatorname {var} (T_{2})}" loading="lazy"></span> for all values of <i>θ</i>. This relationship can be determined by simplifying the more general case above for mean squared error; since the expected value of an unbiased estimator is equal to the parameter value, <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} [T]=\theta }">

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

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</math></span><img src="./149385e4baa32fa775b5c66a9e1da1fa794fa2e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.702ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [T]=\theta }" loading="lazy"></span>. Therefore, for an unbiased estimator, <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 {MSE} (T)=\operatorname {var} (T)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} (T)=\operatorname {var} (T)}</annotation>

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</math></span><img src="./900e60399a972e0d5cc5f674139114e592437589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.297ex; height:2.843ex;" alt="{\displaystyle \operatorname {MSE} (T)=\operatorname {var} (T)}" loading="lazy"></span>, as 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 {E} [T]-\theta )^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle (\operatorname {E} [T]-\theta )^{2}}</annotation>

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</math></span><img src="./65c4e45ecaf2b0048f5f6a0a594a9917d604a607.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.307ex; height:3.176ex;" alt="{\displaystyle (\operatorname {E} [T]-\theta )^{2}}" loading="lazy"></span> term drops out for being equal to 0.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>If an <a href="Estimator_bias" class="mw-redirect" title="Estimator bias">unbiased</a> <a href="Estimator" title="Estimator">estimator</a> of a parameter <i>θ</i> attains <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(T)=1}">

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<annotation encoding="application/x-tex">{\displaystyle e(T)=1}</annotation>

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</math></span><img src="./31e12fe053ee5089f75542c0e7d29eeefab4fc17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.79ex; height:2.843ex;" alt="{\displaystyle e(T)=1}" loading="lazy"></span> for all values of the parameter, then the estimator is called efficient.<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Equivalently, the estimator achieves equality in the <a href="Cram%C3%A9r%E2%80%93Rao_inequality" class="mw-redirect" title="Cramér–Rao inequality">Cramér–Rao inequality</a> for all <i>θ</i>. The <a href="Cram%C3%A9r%E2%80%93Rao_bound" title="Cramér–Rao bound">Cramér–Rao lower bound</a> is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be.

</p><p>An efficient estimator is also the <a href="Minimum_variance_unbiased_estimator" class="mw-redirect" title="Minimum variance unbiased estimator">minimum variance unbiased estimator</a> (MVUE). This is because an efficient estimator maintains equality on the Cramér–Rao inequality for all parameter values, which means it attains the minimum variance for all parameters (the definition of the MVUE). The MVUE estimator, even if it exists, is not necessarily efficient, because "minimum" does not mean equality holds on the Cramér–Rao inequality.

</p><p>Thus an efficient estimator need not exist, but if it does, it is the MVUE.

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<div class="mw-heading mw-heading4"><h4 id="Finite-sample_efficiency">Finite-sample efficiency</h4></div>

<p>Suppose <span class="nowrap">{ <i>P<sub>θ</sub></i> | <i>θ</i> ∈ Θ </span>} is a <a href="Parametric_model" title="Parametric model">parametric model</a> and <span class="nowrap"><i>X</i> = (<i>X</i><sub>1</sub>, …, <i>X<sub>n</sub></i>)</span> are the data sampled from this model. Let <span class="nowrap"><i>T</i> = <i>T</i>(<i>X</i>)</span> be an <a href="Estimator" title="Estimator">estimator</a> for the parameter <i>θ</i>. If this estimator is <a href="Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> (that is, <span class="nowrap">E[ <i>T</i> ] = <i>θ</i></span>), then the <a href="Cram%C3%A9r%E2%80%93Rao_inequality" class="mw-redirect" title="Cramér–Rao inequality">Cramér–Rao inequality</a> states the <a href="Variance" title="Variance">variance</a> of this estimator is bounded from below:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {var} [\,T\,]\ \geq \ {\mathcal {I}}_{\theta }^{-1},}</annotation>

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</math></span><img src="./11af66bdfbc739fb0a6ca2a24e5bfd925441f1b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.805ex; height:3.343ex;" alt="{\displaystyle \operatorname {var} [\,T\,]\ \geq \ {\mathcal {I}}_{\theta }^{-1},}" 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 \scriptstyle {\mathcal {I}}_{\theta }}">

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<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\mathcal {I}}_{\theta }}</annotation>

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</math></span><img src="./e1491e74040d917a435ed68def2fd1e06f65be8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.048ex; width:1.735ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\mathcal {I}}_{\theta }}" loading="lazy"></span> is the <a href="Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information matrix</a> of the model at point <i>θ</i>. Generally, the variance measures the degree of dispersion of a random variable around its mean. Thus estimators with small variances are more concentrated, they estimate the parameters more precisely. We say that the estimator is a <b>finite-sample efficient estimator</b> (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all <span class="nowrap"><i>θ</i> ∈ Θ</span>. Efficient estimators are always <a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">minimum variance unbiased estimators</a>. However the converse is false: There exist point-estimation problems for which the minimum-variance mean-unbiased estimator is inefficient.<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>

</p><p>Historically, finite-sample efficiency was an early optimality criterion. However this criterion has some limitations:

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<ul><li>Finite-sample efficient estimators are extremely rare. In fact, it was proved that efficient estimation is possible only in an <a href="Exponential_family" title="Exponential family">exponential family</a>, and only for the natural parameters of that family.<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></li>

<li>This notion of efficiency is sometimes restricted to the class of <a href="Bias_of_an_estimator" title="Bias of an estimator">unbiased</a> estimators. (Often it is not.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>) Since there are no good theoretical reasons to require that estimators are unbiased, this restriction is inconvenient. In fact, if we use <a href="Mean_squared_error" title="Mean squared error">mean squared error</a> as a selection criterion, many biased estimators will slightly outperform the “best” unbiased ones. For example, in <a href="Multivariate_statistics" title="Multivariate statistics">multivariate statistics</a> for dimension three or more, the mean-unbiased estimator, <a href="Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a>, is <a href="Admissible_procedure" class="mw-redirect" title="Admissible procedure">inadmissible</a>: Regardless of the outcome, its performance is worse than for example the <a href="James%E2%80%93Stein_estimator" title="James–Stein estimator">James–Stein estimator</a>.</li>

<li>Finite-sample efficiency is based on the variance, as a criterion according to which the estimators are judged. A more general approach is to use <a href="Loss_function" title="Loss function">loss functions</a> other than quadratic ones, in which case the finite-sample efficiency can no longer be formulated.</li></ul>

<p>As an example, among the models encountered in practice, efficient estimators exist for: the mean <i>μ</i> of the <a href="Normal_distribution" title="Normal distribution">normal distribution</a> (but not the variance <i>σ</i><sup>2</sup>), parameter <i>λ</i> of the <a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a>, the probability <i>p</i> in the <a href="Binomial_distribution" title="Binomial distribution">binomial</a> or <a href="Multinomial_distribution" title="Multinomial distribution">multinomial distribution</a>.

</p><p>Consider the model of a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> with unknown mean but known variance: <span class="nowrap">{ <i>P<sub>θ</sub></i> = <i>N</i>(<i>θ</i>, <i>σ</i><sup>2</sup>) | <i>θ</i> ∈ <b>R</b> }.</span> The data consists of <i>n</i> <a href="Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> observations from this model: <span class="nowrap"><i>X</i> = (<i>x</i><sub>1</sub>, …, <i>x<sub>n</sub></i>)</span>. We estimate the parameter <i>θ</i> using the <a href="Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> of all observations:

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<annotation encoding="application/x-tex">{\displaystyle T(X)={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\ .}</annotation>

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</math></span><img src="./6aff1054d7203c7360f0cc38ad254e6177f64bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.241ex; height:6.843ex;" alt="{\displaystyle T(X)={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\ .}" loading="lazy"></span></dd></dl>

<p>This estimator has mean <i>θ</i> and variance of <span class="nowrap"><i>σ</i><sup>2</sup> / <i>n</i></span>, which is equal to the reciprocal of the <a href="Fisher_information" title="Fisher information">Fisher information</a> from the sample. Thus, the sample mean is a finite-sample efficient estimator for the mean of the normal distribution.

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<div class="mw-heading mw-heading3"><h3 id="Asymptotic_efficiency">Asymptotic efficiency</h3></div>

<p>Asymptotic efficiency requires <a href="Consistency_(statistics)" title="Consistency (statistics)">Consistency (statistics)</a>, asymptotically normal distribution of the estimator, and an asymptotic variance-covariance matrix no worse than that of any other estimator.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading4"><h4 id="Example:_Median">Example: Median</h4></div>

<p>Consider a sample of size <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 N}">

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> drawn from a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> of mean <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 }">

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

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</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 unit <a href="Variance" title="Variance">variance</a>, i.e., <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 X_{n}\sim {\mathcal {N}}(\mu ,1).}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{n}\sim {\mathcal {N}}(\mu ,1).}</annotation>

</semantics>

</math></span><img src="./126f834fa35a038e8687f0a22ed0ddc92e9fa9c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.57ex; height:3.009ex;" alt="{\displaystyle X_{n}\sim {\mathcal {N}}(\mu ,1).}" loading="lazy"></span>

</p><p>The <a href="Sample_mean" class="mw-redirect" title="Sample mean">sample mean</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 {\overline {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./08fd87d3502c4f9e1b02ac74644474258aeea176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.131ex; height:3.009ex;" alt="{\displaystyle {\overline {X}}}" loading="lazy"></span>, of the sample <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 X_{1},X_{2},\ldots ,X_{N}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{N}}</annotation>

</semantics>

</math></span><img src="./ab39f1a504ce4632876fcd8c0334755705bbe998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.785ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots ,X_{N}}" loading="lazy"></span>, defined as

</p>

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {X}}={\frac {1}{N}}\sum _{n=1}^{N}X_{n}\sim {\mathcal {N}}\left(\mu ,{\frac {1}{N}}\right).}</annotation>

</semantics>

</math></span><img src="./ab58ddec51d1e40fbbb6a45b94db123e874935de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.952ex; height:7.343ex;" alt="{\displaystyle {\overline {X}}={\frac {1}{N}}\sum _{n=1}^{N}X_{n}\sim {\mathcal {N}}\left(\mu ,{\frac {1}{N}}\right).}" loading="lazy"></span></dd></dl>

<p>The variance of the mean, 1/<i>N</i> (the square of the <a href="Standard_error" title="Standard error">standard error</a>) is equal to the reciprocal of the <a href="Fisher_information" title="Fisher information">Fisher information</a> from the sample and thus, by the <a href="Cram%C3%A9r%E2%80%93Rao_inequality" class="mw-redirect" title="Cramér–Rao inequality">Cramér–Rao inequality</a>, the sample mean is efficient in the sense that its efficiency is unity (100%).

</p><p>Now consider the <a href="Sample_median" class="mw-redirect" title="Sample median">sample median</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 {\widetilde {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7333dc05bcf7a18575d85e4d3762ce6f269f21aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.554ex; height:2.676ex;" alt="{\displaystyle {\widetilde {X}}}" loading="lazy"></span>. This is an <a href="Estimator_bias" class="mw-redirect" title="Estimator bias">unbiased</a> and <a href="Consistent_estimator" title="Consistent estimator">consistent</a> estimator for <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 }">

</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>. For large <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 N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> the sample median is approximately <a href="Normal_distribution" title="Normal distribution">normally distributed</a> with mean <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 }">

</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 variance <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 {\pi }/{2N},}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\pi }/{2N},}</annotation>

</semantics>

</math></span><img src="./cd5819eda5a3319390e22534d2b719af1274bc5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.367ex; height:2.843ex;" alt="{\displaystyle {\pi }/{2N},}" loading="lazy"></span><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {X}}\sim {\mathcal {N}}\left(\mu ,{\frac {\pi }{2N}}\right).}</annotation>

</semantics>

</math></span><img src="./5f157f95689998d4ed178436bb3734c71336fc9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.621ex; height:4.843ex;" alt="{\displaystyle {\widetilde {X}}\sim {\mathcal {N}}\left(\mu ,{\frac {\pi }{2N}}\right).}" loading="lazy"></span></dd></dl>

<p>The efficiency of the median for large <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 N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is thus

</p>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\left({\widetilde {X}}\right)=\left({\frac {1}{N}}\right)\left({\frac {\pi }{2N}}\right)^{-1}=2/\pi \approx 0.64.}</annotation>

</semantics>

</math></span><img src="./cbcf2fcd95b671da9a7ddf2771136ae07f687b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.411ex; height:6.176ex;" alt="{\displaystyle e\left({\widetilde {X}}\right)=\left({\frac {1}{N}}\right)\left({\frac {\pi }{2N}}\right)^{-1}=2/\pi \approx 0.64.}" loading="lazy"></span></dd></dl>

<p>In other words, the relative variance of the median will be <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 \pi /2\approx 1.57}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi /2\approx 1.57}</annotation>

</semantics>

</math></span><img src="./4a36ff9e1888aa40c1aded554c531ed6c74ac50f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.89ex; height:2.843ex;" alt="{\displaystyle \pi /2\approx 1.57}" loading="lazy"></span>, or 57% greater than the variance of the mean – the standard error of the median will be 25% greater than that of the mean.<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>

</p><p>Note that this is the <a href="Asymptote" title="Asymptote">asymptotic</a> efficiency — that is, the efficiency in the limit as sample size <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 N}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> tends to infinity. For finite values 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 N,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b2285a1804b7fdcac187d155af09aff63152dd56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle N,}" loading="lazy"></span> the efficiency is higher than this (for example, a sample size of 3 gives an efficiency of about 74%).

</p><p>The sample mean is thus more efficient than the sample median in this example. However, there may be measures by which the median performs better. For example, the median is far more robust to <a href="Outlier" title="Outlier">outliers</a>, so that if the Gaussian model is questionable or approximate, there may advantages to using the median (see <a href="Robust_statistics" title="Robust statistics">Robust statistics</a>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Dominant_estimators">Dominant estimators</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation>

</semantics>

</math></span><img src="./2f304724948a3ef606c4a92459e22b87a954d993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}" 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 T_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation>

</semantics>

</math></span><img src="./d1ba5f12fbb0ff766aec6e22148b429373608555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}" loading="lazy"></span> are estimators for the 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 \theta }">

</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>, 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 T_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation>

</semantics>

</math></span><img src="./2f304724948a3ef606c4a92459e22b87a954d993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}" loading="lazy"></span> is said to <b><a href="Dominating_decision_rule" title="Dominating decision rule">dominate</a></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 T_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation>

</semantics>

</math></span><img src="./d1ba5f12fbb0ff766aec6e22148b429373608555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}" loading="lazy"></span> if:

</p>

<ol><li>its <a href="Mean_squared_error" title="Mean squared error">mean squared error</a> (MSE) is smaller for at least some value 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 \theta }">

</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></li>

<li>the MSE does not exceed that 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 T_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation>

</semantics>

</math></span><img src="./d1ba5f12fbb0ff766aec6e22148b429373608555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}" loading="lazy"></span> for any value of θ.</li></ol>

<p>Formally, <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 T_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation>

</semantics>

</math></span><img src="./2f304724948a3ef606c4a92459e22b87a954d993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}" loading="lazy"></span> dominates <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 T_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation>

</semantics>

</math></span><img src="./d1ba5f12fbb0ff766aec6e22148b429373608555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}" loading="lazy"></span> if

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [(T_{1}-\theta )^{2}]\leq \operatorname {E} [(T_{2}-\theta )^{2}]}</annotation>

</semantics>

</math></span><img src="./a1a6e3b42a820b3e2375ffc6d628bfeeb106d1fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.264ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [(T_{1}-\theta )^{2}]\leq \operatorname {E} [(T_{2}-\theta )^{2}]}" loading="lazy"></span></dd></dl>

<p>holds for all <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 }">

</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>, with strict inequality holding somewhere.

</p>

<div class="mw-heading mw-heading3"><h3 id="Relative_efficiency">Relative efficiency</h3></div>

<p>The relative efficiency of two unbiased estimators is defined as<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>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e(T_{1},T_{2})={\frac {\operatorname {E} [(T_{2}-\theta )^{2}]}{\operatorname {E} [(T_{1}-\theta )^{2}]}}={\frac {\operatorname {var} (T_{2})}{\operatorname {var} (T_{1})}}}</annotation>

</semantics>

</math></span><img src="./19ad6436bd499cfa15f02263dbf3084eec9d45f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.225ex; height:6.676ex;" alt="{\displaystyle e(T_{1},T_{2})={\frac {\operatorname {E} [(T_{2}-\theta )^{2}]}{\operatorname {E} [(T_{1}-\theta )^{2}]}}={\frac {\operatorname {var} (T_{2})}{\operatorname {var} (T_{1})}}}" loading="lazy"></span></dd></dl>

<p>Although <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> is in general 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 \theta }">

</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>, in many cases the dependence drops out; if this is so, <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> being greater than one would indicate 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 T_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation>

</semantics>

</math></span><img src="./2f304724948a3ef606c4a92459e22b87a954d993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}" loading="lazy"></span> is preferable, regardless of the true value 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 \theta }">

</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>.

</p><p>An alternative to relative efficiency for comparing estimators, is the <a href="Pitman_closeness_criterion" title="Pitman closeness criterion">Pitman closeness criterion</a>. This replaces the comparison of mean-squared-errors with comparing how often one estimator produces estimates closer to the true value than another estimator.

</p>

<div class="mw-heading mw-heading4"><h4 id="Estimators_of_the_mean_of_u.i.d._variables">Estimators of the mean of u.i.d. variables</h4></div>

<p>In estimating the mean of uncorrelated, identically distributed variables we can take advantage of the fact that <a href="Variance#Sum_of_uncorrelated_variables_(Bienaymé_formula)" title="Variance">the variance of the sum is the sum of the variances</a>. In this case efficiency can be defined as the square of the <a href="Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a>, i.e.,<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>

</p>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\equiv \left({\frac {\sigma }{\mu }}\right)^{2}}</annotation>

</semantics>

</math></span><img src="./38038996d384c0bebe24ebe2950f89e3ec0213d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.895ex; height:6.509ex;" alt="{\displaystyle e\equiv \left({\frac {\sigma }{\mu }}\right)^{2}}" loading="lazy"></span></dd></dl>

<p>Relative efficiency of two such estimators can thus be interpreted as the relative sample size of one required to achieve the certainty of the other. Proof:

</p>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {s_{1}^{2}}{s_{2}^{2}}}.}</annotation>

</semantics>

</math></span><img src="./c68f9d6bb3b3971b4d61108528da8e435eb20fc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:9.7ex; height:6.843ex;" alt="{\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {s_{1}^{2}}{s_{2}^{2}}}.}" loading="lazy"></span></dd></dl>

<p>Now 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 s_{1}^{2}=n_{1}\sigma ^{2},\,s_{2}^{2}=n_{2}\sigma ^{2}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{1}^{2}=n_{1}\sigma ^{2},\,s_{2}^{2}=n_{2}\sigma ^{2}}</annotation>

</semantics>

</math></span><img src="./d50d090d2c3df55a332d53ca070db2e868db4f42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.575ex; height:3.343ex;" alt="{\displaystyle s_{1}^{2}=n_{1}\sigma ^{2},\,s_{2}^{2}=n_{2}\sigma ^{2}}" loading="lazy"></span> 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 {\frac {e_{1}}{e_{2}}}={\frac {n_{1}}{n_{2}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {n_{1}}{n_{2}}}}</annotation>

</semantics>

</math></span><img src="./4864c8caf73c713d5f1cc6dc9e437ff67d840135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.357ex; height:5.009ex;" alt="{\displaystyle {\frac {e_{1}}{e_{2}}}={\frac {n_{1}}{n_{2}}}}" loading="lazy"></span>, so the relative efficiency expresses the relative sample size of the first estimator needed to match the variance of the second.

</p>

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

<p>Efficiency of an estimator may change significantly if the distribution changes, often dropping. This is one of the motivations of <a href="Robust_statistics" title="Robust statistics">robust statistics</a> – an estimator such as the sample mean is an efficient estimator of the population mean of a normal distribution, for example, but can be an inefficient estimator of a <a href="Mixture_distribution" title="Mixture distribution">mixture distribution</a> of two normal distributions with the same mean and different variances. For example, if a distribution is a combination of 98% <i>N</i>(<i>μ,</i> <i>σ</i>) and 2% <i>N</i>(<i>μ,</i> 10<i>σ</i>), the presence of extreme values from the latter distribution (often "contaminating outliers") significantly reduces the efficiency of the sample mean as an estimator of <i>μ.</i> By contrast, the <a href="Truncated_mean" title="Truncated mean">trimmed mean</a> is less efficient for a normal distribution, but is more robust (i.e., less affected) by changes in the distribution, and thus may be more efficient for a mixture distribution. Similarly, the <a href="Shape_of_a_probability_distribution" title="Shape of a probability distribution">shape of a distribution</a>, such as <a href="Skewness" title="Skewness">skewness</a> or <a href="Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy tails</a>, can significantly reduce the efficiency of estimators that assume a symmetric distribution or thin tails.

</p>

<div class="mw-heading mw-heading3"><h3 id="Efficiency_in_statistics">Efficiency in statistics</h3></div>

<p>Efficiency in statistics is important because it allows the performance of various estimators to be compared. Although an unbiased estimator is usually favored over a biased one, a more efficient biased estimator can sometimes be more valuable than a less efficient unbiased estimator. For example, this can occur when the values of the biased estimator gathers around a number closer to the true value. Thus, estimator performance can be predicted easily by comparing their mean squared errors or variances.

</p>

<div class="mw-heading mw-heading4"><h4 id="Uses_of_inefficient_estimators">Uses of inefficient estimators</h4></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="L-estimator#Applications" title="L-estimator">L-estimator § Applications</a></div>

<p>While efficiency is a desirable quality of an estimator, it must be weighed against other considerations, and an estimator that is efficient for certain distributions may well be inefficient for other distributions. Most significantly, estimators that are efficient for clean data from a simple distribution, such as the normal distribution (which is symmetric, unimodal, and has thin tails) may not be robust to contamination by outliers, and may be inefficient for more complicated distributions. In <a href="Robust_statistics" title="Robust statistics">robust statistics</a>, more importance is placed on robustness and applicability to a wide variety of distributions, rather than efficiency on a single distribution. <a href="M-estimator" title="M-estimator">M-estimators</a> are a general class of estimators motivated by these concerns. They can be designed to yield both robustness and high relative efficiency, though possibly lower efficiency than traditional estimators for some cases. They can be very computationally complicated, however.

</p><p>A more traditional alternative are <a href="L-estimator" title="L-estimator">L-estimators</a>, which are very simple statistics that are easy to compute and interpret, in many cases robust, and often sufficiently efficient for initial estimates. See <a href="L-estimator#Applications" title="L-estimator">applications of L-estimators</a> for further discussion. Inefficient statistics in this sense are discussed in detail in <i>The Atomic Nucleus</i> by R. D. Evans, written before the advent of computers, when efficiently estimating even the arithmetic mean of a sorted series of measurements was laborious.<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>

</p>

<div class="mw-heading mw-heading2"><h2 id="Hypothesis_tests">Hypothesis tests</h2></div>

<p>For comparing <a href="Significance_test" class="mw-redirect" title="Significance test">significance tests</a>, a meaningful measure of efficiency can be defined based on the sample size required for the test to achieve a given task <a href="Statistical_power" class="mw-redirect" title="Statistical power">power</a>.<sup id="cite_ref-FOOTNOTEEveritt2002321_15-0" class="reference"><a href="#cite_note-FOOTNOTEEveritt2002321-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>Pitman efficiency<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> and Bahadur efficiency (or Hodges–Lehmann efficiency)<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><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> relate to the comparison of the performance of <a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">statistical hypothesis testing</a> procedures.

</p>

<div class="mw-heading mw-heading2"><h2 id="Experimental_design">Experimental design</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></div>

<p>For experimental designs, efficiency relates to the ability of a design to achieve the objective of the study with minimal expenditure of resources such as time and money. In simple cases, the relative efficiency of designs can be expressed as the ratio of the sample sizes required to achieve a given objective.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Bayes_estimator" title="Bayes estimator">Bayes estimator</a></li>

<li><a href="Consistent_estimator" title="Consistent estimator">Consistent estimator</a></li>

<li><a href="Hodges'_estimator" title="Hodges' estimator">Hodges' estimator</a></li>

<li><a href="Optimal_instruments" title="Optimal instruments">Optimal instruments</a></li></ul>

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

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<li id="cite_note-FOOTNOTEEveritt2002128-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEEveritt2002128_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEEveritt2002128_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEveritt2002">Everitt 2002</a>, p. 128.</span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFMaindonaldBraun2010" class="citation book cs1">Maindonald, John; Braun, W. John (2010-05-06). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8bMj8m-4RDQC&pg=PA104"><i>Data Analysis and Graphics Using R: An Example-Based Approach</i></a>. Cambridge University Press. p. 104. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-139-48667-5</bdi>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFWackerlyMendenhallScheaffer2008" class="citation book cs1">Wackerly, Dennis D.; Mendenhall, William; Scheaffer, Richard L. (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalstat00wack_691"><i>Mathematical statistics with applications</i></a></span> (Seventh ed.). Belmont, CA: Thomson Brooks/Cole. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalstat00wack_691/page/n469">445</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780495110811</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/183886598">183886598</a>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrubbs1965" class="citation book cs1">Grubbs, Frank (1965). <i>Statistical Measures of Accuracy for Riflemen and Missile Engineers</i>. pp. <span class="nowrap">26–</span>27.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFEvans1955" class="citation book cs1">Evans, Robley D. (1955). <a rel="nofollow" class="external text" href="https://library.psfc.mit.edu/catalog/online_pubs/books/evans_atomic_nucleus.pdf"><i>The Atomic Nucleus</i></a> <span class="cs1-format">(PDF)</span>. McGraw Hill. pp. 746, Appendix G, p902 - Some Useful Inefficient Statistics.</cite></span>

</li>

<li id="cite_note-FOOTNOTEEveritt2002321-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEveritt2002321_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEveritt2002">Everitt 2002</a>, p. 321.</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"><cite id="CITEREFNikitin2001" class="citation cs2">Nikitin, Ya.Yu. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Efficiency,_asymptotic">"Efficiency, asymptotic"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Bahadur_efficiency">"Bahadur efficiency - Encyclopedia of Mathematics"</a>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Arcones M. A. <a rel="nofollow" class="external text" href="http://www.math.binghamton.edu/arcones/prep/pv.pdf">"Bahadur efficiency of the likelihood ratio test"</a> preprint</span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Canay I. A. & Otsu, T. <a rel="nofollow" class="external text" href="http://faculty.wcas.northwestern.edu/~iac879/wp/HL.pdf">"Hodges–Lehmann Optimality for Testing Moment Condition Models"</a></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFDodge2006" class="citation book cs1">Dodge, Y. (2006). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/oxforddictionary0000unse"><i>The Oxford Dictionary of Statistical Terms</i></a></span>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-920613-9</bdi>.</cite></span>

</li>

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

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

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<ul><li><cite id="CITEREFEveritt2002" class="citation book cs1">Everitt, Brian S. (2002). <i>The Cambridge Dictionary of Statistics</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-81099-X</bdi>.</cite></li>

<li><cite id="CITEREFLehmann1998" class="citation book cs1"><a href="Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, Erich L.</a> (1998). <i>Elements of Large-Sample Theory</i>. New York: Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98595-4</bdi>.</cite></li></ul>

</div>

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

<ul><li><cite id="CITEREFLehmannCasella1998" class="citation book cs1"><a href="Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, E.L.</a>; Casella, G. (1998). <i>Theory of Point Estimation</i> (2nd ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98502-6</bdi>.</cite></li>

<li><cite id="CITEREFPfanzaglwith_the_assistance_of_R._Hamböker1994" class="citation book cs1"><a href="Johann_Pfanzagl" title="Johann Pfanzagl">Pfanzagl, Johann</a>; with the assistance of R. Hamböker (1994). <i>Parametric Statistical Theory</i>. Berlin: Walter de Gruyter. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-11-013863-8</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=1291393">1291393</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>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></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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This article is issued from <a class="external text" title="Last edited on 2025-07-17" href="https://en.wikipedia.org/wiki/?title=Efficiency_(statistics)&oldid=1300987313">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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