Bias of an estimator

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<span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Sample_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sample_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Sample variance</span> </div> </a> <ul id="toc-Sample_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estimating_a_Poisson_probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimating_a_Poisson_probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Estimating a Poisson probability</span> </div> </a> <ul id="toc-Estimating_a_Poisson_probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximum_of_a_discrete_uniform_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximum_of_a_discrete_uniform_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Maximum of a discrete uniform distribution</span> </div> </a> <ul id="toc-Maximum_of_a_discrete_uniform_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Median-unbiased_estimators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Median-unbiased_estimators"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Median-unbiased estimators</span> </div> </a> <ul id="toc-Median-unbiased_estimators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bias_with_respect_to_other_loss_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bias_with_respect_to_other_loss_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bias with respect to other loss functions</span> </div> </a> <ul id="toc-Bias_with_respect_to_other_loss_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_of_transformations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Effect_of_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Effect of transformations</span> </div> </a> <ul id="toc-Effect_of_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bias,_variance_and_mean_squared_error" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bias,_variance_and_mean_squared_error"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bias, variance and mean squared error</span> </div> </a> <button aria-controls="toc-Bias,_variance_and_mean_squared_error-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Bias, variance and mean squared error subsection</span> </button> <ul id="toc-Bias,_variance_and_mean_squared_error-sublist" class="vector-toc-list"> <li id="toc-Example:_Estimation_of_population_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_Estimation_of_population_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Example: Estimation of population variance</span> </div> </a> <ul id="toc-Example:_Estimation_of_population_variance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bayesian_view" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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href="https://ca.wikipedia.org/wiki/Biaix_(estad%C3%ADstica)" title="Biaix (estadística) – Catalan" lang="ca" hreflang="ca" data-title="Biaix (estadística)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bias_af_estimator" title="Bias af estimator – Danish" lang="da" hreflang="da" data-title="Bias af estimator" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Verzerrung_einer_Sch%C3%A4tzfunktion" title="Verzerrung einer Schätzfunktion – German" lang="de" hreflang="de" data-title="Verzerrung einer Schätzfunktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sesgo_estad%C3%ADstico" title="Sesgo estadístico – Spanish" lang="es" hreflang="es" data-title="Sesgo estadístico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B1%DB%8C%D8%A8%DB%8C_%DB%8C%DA%A9_%D8%A8%D8%B1%D8%A2%D9%88%D8%B1%D8%AF%DA%AF%D8%B1" title="اریبی یک برآوردگر – Persian" lang="fa" hreflang="fa" data-title="اریبی یک برآوردگر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Biais_(statistique)" title="Biais (statistique) – French" lang="fr" hreflang="fr" data-title="Biais (statistique)" data-language-autonym="Français" 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.hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">Bias (statistics)</a>.</div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>bias of an estimator</b> (or <b>bias function</b>) is the difference between this <a href="/wiki/Estimator" title="Estimator">estimator</a>'s <a href="/wiki/Expected_value" title="Expected value">expected value</a> and the <a href="/wiki/True_value" class="mw-redirect" title="True value">true value</a> of the parameter being estimated. An estimator or decision rule with zero bias is called <i><b>unbiased</b></i>. In statistics, "bias" is an <em>objective</em> property of an estimator. Bias is a distinct concept from <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistency</a>: consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased (see <a href="/wiki/Consistent_estimator#Bias_versus_consistency" title="Consistent estimator">bias versus consistency</a> for more). </p><p>All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in <a href="/wiki/Unbiased_estimation_of_standard_deviation" title="Unbiased estimation of standard deviation">unbiased estimation of standard deviation</a>); because a biased estimator may be unbiased with respect to different measures of <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a>; because a biased estimator gives a lower value of some <a href="/wiki/Loss_function" title="Loss function">loss function</a> (particularly <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a>) compared with unbiased estimators (notably in <a href="/wiki/Shrinkage_estimator" class="mw-redirect" title="Shrinkage estimator">shrinkage estimators</a>); or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful. </p><p>Bias can also be measured with respect to the <a href="/wiki/Median" title="Median">median</a>, rather than the mean (expected value), in which case one distinguishes <i>median</i>-unbiased from the usual <i>mean</i>-unbiasedness property. Mean-unbiasedness is not preserved under non-linear <a href="/wiki/Data_transformation_(statistics)" title="Data transformation (statistics)">transformations</a>, though median-unbiasedness is (see <a href="#Effect_of_transformations">§ Effect of transformations</a>); for example, the <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> is a biased estimator for the population variance. These are all illustrated below. </p><p>An unbiased estimator for a parameter need not always exist. For example, there is no unbiased estimator for the reciprocal of the parameter of a binomial random variable.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose we have a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a>, parameterized by a real number <i>θ</i>, giving rise to a probability distribution for observed data, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\theta }(x)=P(x\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\theta }(x)=P(x\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0c92a2f9f331565f351f716b59a2d1167eebbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.645ex; height:2.843ex;" alt="{\displaystyle P_{\theta }(x)=P(x\mid \theta )}"></span>, and a statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}"></span> which serves as an <a href="/wiki/Estimator" title="Estimator">estimator</a> of <i>θ</i> based on any observed data <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. That is, we assume that our data follows some unknown distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88296f80b46f26a18ff8e11652dd7ca556f2fb8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle P(x\mid \theta )}"></span> (where <i>θ</i> is a fixed, unknown constant that is part of this distribution), and then we construct some estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}"></span> that maps observed data to values that we hope are close to <i>θ</i>. The <b>bias</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}"></span> relative to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" 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 }"></span> is defined as<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><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 {Bias} ({\hat {\theta }},\theta )=\operatorname {Bias} _{\theta }[\,{\hat {\theta }}\,]=\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}\,]-\theta =\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}-\theta \,],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Bias</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Bias</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>−</mo> <mi>θ</mi> <mo>=</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Bias} ({\hat {\theta }},\theta )=\operatorname {Bias} _{\theta }[\,{\hat {\theta }}\,]=\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}\,]-\theta =\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}-\theta \,],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079cf00ac87392228fa3bcf23d37b26a8d83d7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.078ex; height:3.676ex;" alt="{\displaystyle \operatorname {Bias} ({\hat {\theta }},\theta )=\operatorname {Bias} _{\theta }[\,{\hat {\theta }}\,]=\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}\,]-\theta =\operatorname {E} _{x\mid \theta }[\,{\hat {\theta }}-\theta \,],}"></span></dd></dl> <p>where <span class="mwe-math-element"><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} _{x\mid \theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{x\mid \theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54ff58946b27eaddcb35b7c337af8cbc4be8efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.984ex; height:3.009ex;" alt="{\displaystyle \operatorname {E} _{x\mid \theta }}"></span> denotes <a href="/wiki/Expected_value" title="Expected value">expected value</a> over the distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88296f80b46f26a18ff8e11652dd7ca556f2fb8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle P(x\mid \theta )}"></span> (i.e., averaging over all possible observations <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>). The second equation follows since <i>θ</i> is measurable with respect to the conditional distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x\mid \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x\mid \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88296f80b46f26a18ff8e11652dd7ca556f2fb8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle P(x\mid \theta )}"></span>. </p><p>An estimator is said to be <b>unbiased</b> if its bias is equal to zero for all values of parameter <i>θ</i>, or equivalently, if the expected value of the estimator matches that of the parameter.<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> Unbiasedness is not guaranteed to carry over. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}"></span> is an unbiased estimator for parameter <i>θ</i>, it is not guaranteed that g(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eaae56d74c5844e86caeed8ae205ff9e413bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\hat {\theta }}}"></span>) is an unbiased estimator for g(<i>θ).</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the <a href="/wiki/Mean_signed_difference" class="mw-redirect" title="Mean signed difference">mean signed difference</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sample_variance">Sample variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=3" title="Edit section: Sample variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">Sample variance</a></div> <p>The <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> of a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms of <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a> (MSE), which can be minimized by using a different scale factor, resulting in a biased estimator with lower MSE than the unbiased estimator. Concretely, the naive estimator sums the squared deviations and divides by <i>n,</i> which is biased. Dividing instead by <i>n</i> − 1 yields an unbiased estimator. Conversely, MSE can be minimized by dividing by a different number (depending on distribution), but this results in a biased estimator. This number is always larger than <i>n</i> − 1, so this is known as a <a href="/wiki/Shrinkage_estimator" class="mw-redirect" title="Shrinkage estimator">shrinkage estimator</a>, as it "shrinks" the unbiased estimator towards zero; for the normal distribution the optimal value is <i>n</i> + 1. </p><p>Suppose <i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub> are <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> (i.i.d.) random variables with <a href="/wiki/Expected_value" title="Expected value">expectation</a> <i>μ</i> and <a href="/wiki/Variance" title="Variance">variance</a> <i>σ</i><sup>2</sup>. If the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> and uncorrected <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> are defined as </p> <dl><dd><span class="mwe-math-element"><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}}\,={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\qquad S^{2}={\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}\,{\big )}^{2}\qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="2em" /> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}\,={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\qquad S^{2}={\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}\,{\big )}^{2}\qquad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329507ab4567c66e5d24da60a5b4954333eb2bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.292ex; height:6.843ex;" alt="{\displaystyle {\overline {X}}\,={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\qquad S^{2}={\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}\,{\big )}^{2}\qquad }"></span></dd></dl> <p>then <i>S</i><sup>2</sup> is a biased estimator of <i>σ</i><sup>2</sup>, because </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )-({\overline {X}}-\mu ){\bigg )}^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg )}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\sum _{i=1}^{n}1{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\cdot n{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )-({\overline {X}}-\mu ){\bigg )}^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg )}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\sum _{i=1}^{n}1{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\cdot n{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06f42fbaf9fe45412f5dab8d48bc953133132aa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.171ex; width:74.901ex; height:43.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )-({\overline {X}}-\mu ){\bigg )}^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}{\bigg (}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg )}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\sum _{i=1}^{n}1{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+{\frac {1}{n}}({\overline {X}}-\mu )^{2}\cdot n{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]\end{aligned}}}"></span></dd></dl> <p>To continue, we note that by subtracting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" 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 }"></span> from both sides of <span class="mwe-math-element"><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}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bba8c8160cf89fc97efe65f1c8c47126eded5c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.314ex; height:6.843ex;" alt="{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}"></span>, we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\overline {X}}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-{\frac {1}{n}}\sum _{i=1}^{n}\mu \ ={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu ).\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>μ</mi> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {X}}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-{\frac {1}{n}}\sum _{i=1}^{n}\mu \ ={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu ).\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312e2112436605a166b8301e4914e242c20077e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:65.408ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {X}}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-\mu ={\frac {1}{n}}\sum _{i=1}^{n}X_{i}-{\frac {1}{n}}\sum _{i=1}^{n}\mu \ ={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu ).\\[8pt]\end{aligned}}}"></span></dd></dl> <p>Meaning, (by cross-multiplication) <span class="mwe-math-element"><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\cdot ({\overline {X}}-\mu )=\sum _{i=1}^{n}(X_{i}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot ({\overline {X}}-\mu )=\sum _{i=1}^{n}(X_{i}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625e19be8c950287ad48fbc2f37eceb9d18d3392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.485ex; height:6.843ex;" alt="{\displaystyle n\cdot ({\overline {X}}-\mu )=\sum _{i=1}^{n}(X_{i}-\mu )}"></span>. Then, the previous becomes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\cdot n\cdot ({\overline {X}}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )^{2}+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\sigma ^{2}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}=\left(1-{\frac {1}{n}}\right)\sigma ^{2}<\sigma ^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\cdot n\cdot ({\overline {X}}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )^{2}+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\sigma ^{2}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}=\left(1-{\frac {1}{n}}\right)\sigma ^{2}<\sigma ^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f77971bd3f08e0c66f208221059b6978ab507e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.838ex; width:67.796ex; height:50.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\sum _{i=1}^{n}(X_{i}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-{\frac {2}{n}}({\overline {X}}-\mu )\cdot n\cdot ({\overline {X}}-\mu )+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-2({\overline {X}}-\mu )^{2}+({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}-({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}\\[8pt]&=\sigma ^{2}-\operatorname {E} {\bigg [}({\overline {X}}-\mu )^{2}{\bigg ]}=\left(1-{\frac {1}{n}}\right)\sigma ^{2}<\sigma ^{2}.\end{aligned}}}"></span></dd></dl> <p>This can be seen by noting the following formula, which follows from the <a href="/wiki/Variance#Sum_of_uncorrelated_variables" title="Variance">Bienaymé formula</a>, for the term in the inequality for the expectation of the uncorrected sample variance above: <span class="mwe-math-element"><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} {\big [}({\overline {X}}-\mu )^{2}{\big ]}={\frac {1}{n}}\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}={\frac {1}{n}}\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d386a26eb18f6db56ca2f12646c2484c0e35cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.86ex; height:5.176ex;" alt="{\displaystyle \operatorname {E} {\big [}({\overline {X}}-\mu )^{2}{\big ]}={\frac {1}{n}}\sigma ^{2}}"></span>. </p><p>In other words, the expected value of the uncorrected sample variance does not equal the population variance <i>σ</i><sup>2</sup>, unless multiplied by a normalization factor. The sample mean, on the other hand, is an unbiased<sup id="cite_ref-JohnsonWichern2007_5-0" class="reference"><a href="#cite_note-JohnsonWichern2007-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> estimator of the population mean <i>μ</i>.<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>Note that the usual definition of sample variance is <span class="mwe-math-element"><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^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3c0061ec4b0b3aa8222732875dba29a799c0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.596ex; height:6.843ex;" alt="{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}"></span>, and this is an unbiased estimator of the population variance. </p><p>Algebraically speaking, <span class="mwe-math-element"><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} [S^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [S^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270b0ddfeee20bb27511ac518d98ef767c813bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.452ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [S^{2}]}"></span> is unbiased because: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n-1}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]={\frac {n}{n-1}}\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]\\[8pt]&={\frac {n}{n-1}}\left(1-{\frac {1}{n}}\right)\sigma ^{2}=\sigma ^{2},\\[8pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n-1}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]={\frac {n}{n-1}}\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]\\[8pt]&={\frac {n}{n-1}}\left(1-{\frac {1}{n}}\right)\sigma ^{2}=\sigma ^{2},\\[8pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625ed97f0fa7deca979cf3c92750130c003df8f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:66.09ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} \left[{\frac {1}{n-1}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]={\frac {n}{n-1}}\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\big (}X_{i}-{\overline {X}}{\big )}^{2}\right]\\[8pt]&={\frac {n}{n-1}}\left(1-{\frac {1}{n}}\right)\sigma ^{2}=\sigma ^{2},\\[8pt]\end{aligned}}}"></span></dd></dl> <p>where the transition to the second line uses the result derived above for the biased estimator. Thus <span class="mwe-math-element"><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} [S^{2}]=\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [S^{2}]=\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b991e76d21d27a2eaa961002c06c4722174a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.936ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [S^{2}]=\sigma ^{2}}"></span>, and therefore <span class="mwe-math-element"><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^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3c0061ec4b0b3aa8222732875dba29a799c0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.596ex; height:6.843ex;" alt="{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}}\,)^{2}}"></span> is an unbiased estimator of the population variance, <i>σ</i><sup>2</sup>. The ratio between the biased (uncorrected) and unbiased estimates of the variance is known as <a href="/wiki/Bessel%27s_correction" title="Bessel's correction">Bessel's correction</a>. </p><p>The reason that an uncorrected sample variance, <i>S</i><sup>2</sup>, is biased stems from the fact that the sample mean is an <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> (OLS) estimator for <i>μ</i>: <span class="mwe-math-element"><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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08fd87d3502c4f9e1b02ac74644474258aeea176" 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}}}"></span> is the number that makes the sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b64f61ebac7211f860efd1f85c9c82dc627f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.914ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}"></span> as small as possible. That is, when any other number is plugged into this sum, the sum can only increase. In particular, the choice <span class="mwe-math-element"><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 \neq {\overline {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \neq {\overline {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30f4d89d69d22550b9571486822bd3f2c64703f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:3.509ex;" alt="{\displaystyle \mu \neq {\overline {X}}}"></span> gives, </p> <dl><dd><span class="mwe-math-element"><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 {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}<{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}<{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b519da3367d2ce4b6ae04c3f28544e7dd7b47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.08ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}<{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2},}"></span></dd></dl> <p>and then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}{\bigg ]}<\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}=\sigma ^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo><</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}{\bigg ]}<\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}=\sigma ^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e493cb505bfe33759adff20c05469dfd02718e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.715ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [S^{2}]&=\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}{\bigg ]}<\operatorname {E} {\bigg [}{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-\mu )^{2}{\bigg ]}=\sigma ^{2}.\end{aligned}}}"></span></dd></dl> <p>The above discussion can be understood in geometric terms: the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {C}}=(X_{1}-\mu ,\ldots ,X_{n}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {C}}=(X_{1}-\mu ,\ldots ,X_{n}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8534d13fe24457f149a14df3aaf5c5a20833f9cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.458ex; height:3.509ex;" alt="{\displaystyle {\vec {C}}=(X_{1}-\mu ,\ldots ,X_{n}-\mu )}"></span> can be decomposed into the "mean part" and "variance part" by projecting to the direction of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}=(1,\ldots ,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}=(1,\ldots ,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9848e263616f2d8adf360182b23237f57775b3b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.741ex; height:2.843ex;" alt="{\displaystyle {\vec {u}}=(1,\ldots ,1)}"></span> and to that direction's orthogonal complement hyperplane. One gets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}}=({\overline {X}}-\mu ,\ldots ,{\overline {X}}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}=({\overline {X}}-\mu ,\ldots ,{\overline {X}}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/288888b04cda1f4babdefb9c92f33a23a367347b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.576ex; height:3.509ex;" alt="{\displaystyle {\vec {A}}=({\overline {X}}-\mu ,\ldots ,{\overline {X}}-\mu )}"></span> for the part along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}=(X_{1}-{\overline {X}},\ldots ,X_{n}-{\overline {X}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}=(X_{1}-{\overline {X}},\ldots ,X_{n}-{\overline {X}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669bd91852f23d851d8245c9f504e8b36f29bc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.915ex; height:3.509ex;" alt="{\displaystyle {\vec {B}}=(X_{1}-{\overline {X}},\ldots ,X_{n}-{\overline {X}})}"></span> for the complementary part. Since this is an orthogonal decomposition, Pythagorean theorem says <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {C}}|^{2}=|{\vec {A}}|^{2}+|{\vec {B}}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {C}}|^{2}=|{\vec {A}}|^{2}+|{\vec {B}}|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebd7b4c909f3bf01768b628a96dd6f17569dc9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.256ex; height:3.509ex;" alt="{\displaystyle |{\vec {C}}|^{2}=|{\vec {A}}|^{2}+|{\vec {B}}|^{2}}"></span>, and taking expectations we get <span class="mwe-math-element"><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\sigma ^{2}=n\operatorname {E} \left[({\overline {X}}-\mu )^{2}\right]+n\operatorname {E} [S^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>n</mi> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>n</mi> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\sigma ^{2}=n\operatorname {E} \left[({\overline {X}}-\mu )^{2}\right]+n\operatorname {E} [S^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9909a909c3cc268489eb7372c05ef3c7ccc9a07a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.749ex; height:4.843ex;" alt="{\displaystyle n\sigma ^{2}=n\operatorname {E} \left[({\overline {X}}-\mu )^{2}\right]+n\operatorname {E} [S^{2}]}"></span>, as above (but times <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>). If the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757b18b7508788dc438a475d6b5868479de53d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:3.009ex;" alt="{\displaystyle {\vec {C}}}"></span> is rotationally symmetric, as in the case when <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> are sampled from a Gaussian, then on average, the dimension along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> contributes to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {C}}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {C}}|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8db69e618ab9d0dc38cc2ad5925648b5c6623d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.114ex; height:3.509ex;" alt="{\displaystyle |{\vec {C}}|^{2}}"></span> equally as the <span class="mwe-math-element"><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-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> directions perpendicular to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span>, so that <span class="mwe-math-element"><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} \left[({\overline {X}}-\mu )^{2}\right]={\frac {\sigma ^{2}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[({\overline {X}}-\mu )^{2}\right]={\frac {\sigma ^{2}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c9b13c46f7a9c6b5ed2f0d61993a82721c4333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.334ex; height:5.676ex;" alt="{\displaystyle \operatorname {E} \left[({\overline {X}}-\mu )^{2}\right]={\frac {\sigma ^{2}}{n}}}"></span> and <span class="mwe-math-element"><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} [S^{2}]={\frac {(n-1)\sigma ^{2}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [S^{2}]={\frac {(n-1)\sigma ^{2}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672a2045b33e88d6199d8d2ec4bfb3627629902a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.979ex; height:5.843ex;" alt="{\displaystyle \operatorname {E} [S^{2}]={\frac {(n-1)\sigma ^{2}}{n}}}"></span>. This is in fact true in general, as explained above. </p> <div class="mw-heading mw-heading3"><h3 id="Estimating_a_Poisson_probability">Estimating a Poisson probability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=4" title="Edit section: Estimating a Poisson probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A far more extreme case of a biased estimator being better than any unbiased estimator arises from the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Suppose that <i>X</i> has a Poisson distribution with expectation <i>λ</i>. Suppose it is desired to estimate </p> <dl><dd><span class="mwe-math-element"><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 {P} (X=0)^{2}=e^{-2\lambda }\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>λ</mi> </mrow> </msup> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X=0)^{2}=e^{-2\lambda }\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e958a140fcb53739d20984fde752bf20b49a27c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.483ex; height:3.176ex;" alt="{\displaystyle \operatorname {P} (X=0)^{2}=e^{-2\lambda }\quad }"></span></dd></dl> <p>with a sample of size 1. (For example, when incoming calls at a telephone switchboard are modeled as a Poisson process, and <i>λ</i> is the average number of calls per minute, then <i>e</i><sup>−2<i>λ</i></sup> is the probability that no calls arrive in the next two minutes.) </p><p>Since the expectation of an unbiased estimator <i>δ</i>(<i>X</i>) is equal to the <a href="/wiki/Estimand" title="Estimand">estimand</a>, i.e. </p> <dl><dd><span class="mwe-math-element"><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} (\delta (X))=\sum _{x=0}^{\infty }\delta (x){\frac {\lambda ^{x}e^{-\lambda }}{x!}}=e^{-2\lambda },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>x</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>λ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\delta (X))=\sum _{x=0}^{\infty }\delta (x){\frac {\lambda ^{x}e^{-\lambda }}{x!}}=e^{-2\lambda },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e7cb29c6615062ae7f7a583846e011cc4765bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.295ex; height:6.843ex;" alt="{\displaystyle \operatorname {E} (\delta (X))=\sum _{x=0}^{\infty }\delta (x){\frac {\lambda ^{x}e^{-\lambda }}{x!}}=e^{-2\lambda },}"></span></dd></dl> <p>the only function of the data constituting an unbiased estimator is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)=(-1)^{x}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)=(-1)^{x}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d21529a2d66d1ecd92ac985ff458c3633343e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.272ex; height:2.843ex;" alt="{\displaystyle \delta (x)=(-1)^{x}.\,}"></span></dd></dl> <p>To see this, note that when decomposing e<sup>−<i>λ</i></sup> from the above expression for expectation, the sum that is left is a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion of e<sup>−<i>λ</i></sup> as well, yielding e<sup>−<i>λ</i></sup>e<sup>−<i>λ</i></sup> = e<sup>−2<i>λ</i></sup> (see <a href="/wiki/Characterizations_of_the_exponential_function" title="Characterizations of the exponential function">Characterizations of the exponential function</a>). </p><p>If the observed value of <i>X</i> is 100, then the estimate is 1, although the true value of the quantity being estimated is very likely to be near 0, which is the opposite extreme. And, if <i>X</i> is observed to be 101, then the estimate is even more absurd: It is −1, although the quantity being estimated must be positive. </p><p>The (biased) <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood estimator</a> </p> <dl><dd><span class="mwe-math-element"><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^{-2{X}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </mrow> </msup> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-2{X}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b0d44e3cd1d24fb887a4884f146760cc3a5631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.139ex; height:2.676ex;" alt="{\displaystyle e^{-2{X}}\quad }"></span></dd></dl> <p>is far better than this unbiased estimator. Not only is its value always positive but it is also more accurate in the sense that its <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a> </p> <dl><dd><span class="mwe-math-element"><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^{-4\lambda }-2e^{\lambda (1/e^{2}-3)}+e^{\lambda (1/e^{4}-1)}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> <mi>λ</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-4\lambda }-2e^{\lambda (1/e^{2}-3)}+e^{\lambda (1/e^{4}-1)}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef5d7d50729887786b33f2d2d71abda188b342a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.396ex; height:3.176ex;" alt="{\displaystyle e^{-4\lambda }-2e^{\lambda (1/e^{2}-3)}+e^{\lambda (1/e^{4}-1)}\,}"></span></dd></dl> <p>is smaller; compare the unbiased estimator's MSE of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-e^{-4\lambda }.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> <mi>λ</mi> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-e^{-4\lambda }.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f597159d19d1094a821414fdb48cf544852073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.411ex; height:2.843ex;" alt="{\displaystyle 1-e^{-4\lambda }.\,}"></span></dd></dl> <p>The MSEs are functions of the true value <i>λ</i>. The bias of the maximum-likelihood estimator is: </p> <dl><dd><span class="mwe-math-element"><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^{-2\lambda }-e^{\lambda (1/e^{2}-1)}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>λ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-2\lambda }-e^{\lambda (1/e^{2}-1)}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c56a9c5cfdba92f5489d973f7b344a1895287c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.144ex; height:3.176ex;" alt="{\displaystyle e^{-2\lambda }-e^{\lambda (1/e^{2}-1)}.\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Maximum_of_a_discrete_uniform_distribution">Maximum of a discrete uniform distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=5" title="Edit section: Maximum of a discrete uniform distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Maximum_of_a_discrete_uniform_distribution" class="mw-redirect" title="Maximum of a discrete uniform distribution">Maximum of a discrete uniform distribution</a></div> <p>The bias of maximum-likelihood estimators can be substantial. Consider a case where <i>n</i> tickets numbered from 1 to <i>n</i> are placed in a box and one is selected at random, giving a value <i>X</i>. If <i>n</i> is unknown, then the maximum-likelihood estimator of <i>n</i> is <i>X</i>, even though the expectation of <i>X</i> given <i>n</i> is only (<i>n</i> + 1)/2; we can be certain only that <i>n</i> is at least <i>X</i> and is probably more. In this case, the natural unbiased estimator is 2<i>X</i> − 1. </p> <div class="mw-heading mw-heading2"><h2 id="Median-unbiased_estimators">Median-unbiased estimators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=6" title="Edit section: Median-unbiased estimators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theory of <a href="/wiki/Median" title="Median">median</a>-unbiased estimators was revived by George W. Brown in 1947:<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> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>An estimate of a one-dimensional parameter θ will be said to be median-unbiased, if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.</p></blockquote> <p>Further properties of median-unbiased estimators have been noted by Lehmann, Birnbaum, van der Vaart and Pfanzagl.<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> In particular, median-unbiased estimators exist in cases where mean-unbiased and <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum-likelihood</a> estimators do not exist. They are invariant under <a href="/wiki/Injective_function" title="Injective function">one-to-one transformations</a>. </p><p>There are methods of construction median-unbiased estimators for probability distributions that have <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio">monotone likelihood-functions</a>, such as one-parameter exponential families, to ensure that they are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators).<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><sup id="cite_ref-BrownEtAl_11-0" class="reference"><a href="#cite_note-BrownEtAl-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao–Blackwell procedure for mean-unbiased estimation but for a larger class of loss-functions.<sup id="cite_ref-BrownEtAl_11-1" class="reference"><a href="#cite_note-BrownEtAl-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Bias_with_respect_to_other_loss_functions">Bias with respect to other loss functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=7" title="Edit section: Bias with respect to other loss functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any minimum-variance <i>mean</i>-unbiased estimator minimizes the <a href="/wiki/Risk_(statistics)" class="mw-redirect" title="Risk (statistics)">risk</a> (<a href="/wiki/Expected_loss" title="Expected loss">expected loss</a>) with respect to the squared-error <a href="/wiki/Loss_function" title="Loss function">loss function</a> (among mean-unbiased estimators), as observed by <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>.<sup id="cite_ref-Dodge_12-0" class="reference"><a href="#cite_note-Dodge-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A minimum-<a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">average absolute deviation</a> <i>median</i>-unbiased estimator minimizes the risk with respect to the <a href="/wiki/Absolute_value" title="Absolute value">absolute</a> loss function (among median-unbiased estimators), as observed by <a href="/wiki/Laplace" class="mw-redirect" title="Laplace">Laplace</a>.<sup id="cite_ref-Dodge_12-1" class="reference"><a href="#cite_note-Dodge-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Other loss functions are used in statistics, particularly in <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistics</a>.<sup id="cite_ref-Dodge_12-2" class="reference"><a href="#cite_note-Dodge-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effect_of_transformations">Effect of transformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=8" title="Edit section: Effect of transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For univariate parameters, median-unbiased estimators remain median-unbiased under <a href="/wiki/Data_transformation_(statistics)" title="Data transformation (statistics)">transformations</a> that preserve order (or reverse order). Note that, when a transformation is applied to a mean-unbiased estimator, the result need not be a mean-unbiased estimator of its corresponding population statistic. By <a href="/wiki/Jensen%27s_inequality" title="Jensen's inequality">Jensen's inequality</a>, a <a href="/wiki/Convex_function" title="Convex function">convex function</a> as transformation will introduce positive bias, while a <a href="/wiki/Concave_function" title="Concave function">concave function</a> will introduce negative bias, and a function of mixed convexity may introduce bias in either direction, depending on the specific function and distribution. That is, for a non-linear function <i>f</i> and a mean-unbiased estimator <i>U</i> of a parameter <i>p</i>, the composite estimator <i>f</i>(<i>U</i>) need not be a mean-unbiased estimator of <i>f</i>(<i>p</i>). For example, the <a href="/wiki/Square_root" title="Square root">square root</a> of the unbiased estimator of the population <a href="/wiki/Variance" title="Variance">variance</a> is <em>not</em> a mean-unbiased estimator of the population <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>: the square root of the unbiased <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a>, the corrected <a href="/wiki/Sample_standard_deviation" class="mw-redirect" title="Sample standard deviation">sample standard deviation</a>, is biased. The bias depends both on the sampling distribution of the estimator and on the transform, and can be quite involved to calculate – see <a href="/wiki/Unbiased_estimation_of_standard_deviation" title="Unbiased estimation of standard deviation">unbiased estimation of standard deviation</a> for a discussion in this case. </p> <div class="mw-heading mw-heading2"><h2 id="Bias,_variance_and_mean_squared_error"><span id="Bias.2C_variance_and_mean_squared_error"></span>Bias, variance and mean squared error</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=9" title="Edit section: Bias, variance and mean squared error"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Accuracy_(trueness_and_precision)" class="mw-redirect" title="Accuracy (trueness and precision)">Accuracy (trueness and precision)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Example_when_estimator_bias_is_good.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Example_when_estimator_bias_is_good.svg/220px-Example_when_estimator_bias_is_good.svg.png" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Example_when_estimator_bias_is_good.svg/330px-Example_when_estimator_bias_is_good.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Example_when_estimator_bias_is_good.svg/440px-Example_when_estimator_bias_is_good.svg.png 2x" data-file-width="180" data-file-height="121" /></a><figcaption>Sampling distributions of two alternative estimators for a parameter β<sub>0</sub>. Although β<sub>1</sub><sup style="position:relative;left:-8pt;top:-1pt;">^</sup> is unbiased, it is clearly inferior to the biased β<sub>2</sub><sup style="position:relative;left:-8pt;top:-1pt;">^</sup>.<br /><br /><a href="/wiki/Ridge_regression" title="Ridge regression">Ridge regression</a> is one example of a technique where allowing a little bias may lead to a considerable reduction in variance, and more reliable estimates overall.</figcaption></figure> <p>While bias quantifies the <i>average</i> difference to be expected between an estimator and an underlying parameter, an estimator based on a finite sample can additionally be expected to differ from the parameter due to the randomness in the sample. An estimator that minimises the bias will not necessarily minimise the mean square error. One measure which is used to try to reflect both types of difference is the <a href="/wiki/Mean_square_error" class="mw-redirect" title="Mean square error">mean square error</a>,<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><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} ({\hat {\theta }})=\operatorname {E} {\big [}({\hat {\theta }}-\theta )^{2}{\big ]}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>MSE</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {E} {\big [}({\hat {\theta }}-\theta )^{2}{\big ]}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49caa6db3ae779d2a0986756d13c3dc7979a39c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.976ex; height:3.509ex;" alt="{\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {E} {\big [}({\hat {\theta }}-\theta )^{2}{\big ]}.}"></span></dd></dl> <p>This can be shown to be equal to the square of the bias, plus the variance:<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})=&(\operatorname {E} [{\hat {\theta }}]-\theta )^{2}+\operatorname {E} [\,({\hat {\theta }}-\operatorname {E} [\,{\hat {\theta }}\,])^{2}\,]\\=&(\operatorname {Bias} ({\hat {\theta }},\theta ))^{2}+\operatorname {Var} ({\hat {\theta }})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>MSE</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>−</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>Bias</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})=&(\operatorname {E} [{\hat {\theta }}]-\theta )^{2}+\operatorname {E} [\,({\hat {\theta }}-\operatorname {E} [\,{\hat {\theta }}\,])^{2}\,]\\=&(\operatorname {Bias} ({\hat {\theta }},\theta ))^{2}+\operatorname {Var} ({\hat {\theta }})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b541af7bcf5ddc8207f44598b2041e41cfee5544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.961ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})=&(\operatorname {E} [{\hat {\theta }}]-\theta )^{2}+\operatorname {E} [\,({\hat {\theta }}-\operatorname {E} [\,{\hat {\theta }}\,])^{2}\,]\\=&(\operatorname {Bias} ({\hat {\theta }},\theta ))^{2}+\operatorname {Var} ({\hat {\theta }})\end{aligned}}}"></span></dd></dl> <p>When the parameter is a vector, an analogous decomposition applies:<sup id="cite_ref-taboga_15-0" class="reference"><a href="#cite_note-taboga-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><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} ({\hat {\theta }})=\operatorname {trace} (\operatorname {Cov} ({\hat {\theta }}))+\left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>MSE</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>trace</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>Bias</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {trace} (\operatorname {Cov} ({\hat {\theta }}))+\left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/904da260bc7e54a1b6605af749dc9bcaa96a391d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.236ex; height:4.343ex;" alt="{\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {trace} (\operatorname {Cov} ({\hat {\theta }}))+\left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}"></span></dd></dl> <p>where <span class="mwe-math-element"><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 {trace} (\operatorname {Cov} ({\hat {\theta }}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>trace</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {trace} (\operatorname {Cov} ({\hat {\theta }}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a410df179cab2ec290e1dec0f6f1b78779b43179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.086ex; height:3.343ex;" alt="{\displaystyle \operatorname {trace} (\operatorname {Cov} ({\hat {\theta }}))}"></span> is the trace (diagonal sum) of the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> of the estimator and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>Bias</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a55b763f0d0fde9260777eab7aca1dc4c84b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.04ex; height:4.343ex;" alt="{\displaystyle \left\Vert \operatorname {Bias} ({\hat {\theta }},\theta )\right\Vert ^{2}}"></span> is the square <a href="/wiki/Vector_norm" class="mw-redirect" title="Vector norm">vector norm</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_Estimation_of_population_variance">Example: Estimation of population variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=10" title="Edit section: Example: Estimation of population variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For example,<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> suppose an estimator of the form </p> <dl><dd><span class="mwe-math-element"><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}=c\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\,\right)^{2}=cnS^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>c</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>n</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}=c\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\,\right)^{2}=cnS^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f79bc1f6e94cea6421ea4252b6dc2199c864bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.997ex; height:6.843ex;" alt="{\displaystyle T^{2}=c\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}\,\right)^{2}=cnS^{2}}"></span></dd></dl> <p>is sought for the population variance as above, but this time to minimise the MSE: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {MSE} =&\operatorname {E} \left[(T^{2}-\sigma ^{2})^{2}\right]\\=&\left(\operatorname {E} \left[T^{2}-\sigma ^{2}\right]\right)^{2}+\operatorname {Var} (T^{2})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>MSE</mi> <mo>=</mo> </mtd> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {MSE} =&\operatorname {E} \left[(T^{2}-\sigma ^{2})^{2}\right]\\=&\left(\operatorname {E} \left[T^{2}-\sigma ^{2}\right]\right)^{2}+\operatorname {Var} (T^{2})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768d4c44328a35d581ce47e6227b7f1403e180ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.867ex; margin-bottom: -0.304ex; width:34.158ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {MSE} =&\operatorname {E} \left[(T^{2}-\sigma ^{2})^{2}\right]\\=&\left(\operatorname {E} \left[T^{2}-\sigma ^{2}\right]\right)^{2}+\operatorname {Var} (T^{2})\end{aligned}}}"></span></dd></dl> <p>If the variables <i>X</i><sub>1</sub> ... <i>X</i><sub><i>n</i></sub> follow a normal distribution, then <i>nS</i><sup>2</sup>/σ<sup>2</sup> has a <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <i>n</i> − 1 degrees of freedom, giving: </p> <dl><dd><span class="mwe-math-element"><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} [nS^{2}]=(n-1)\sigma ^{2}{\text{ and }}\operatorname {Var} (nS^{2})=2(n-1)\sigma ^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>n</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [nS^{2}]=(n-1)\sigma ^{2}{\text{ and }}\operatorname {Var} (nS^{2})=2(n-1)\sigma ^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95726bf68b25a238b51a7301918165ee854ef732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.93ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [nS^{2}]=(n-1)\sigma ^{2}{\text{ and }}\operatorname {Var} (nS^{2})=2(n-1)\sigma ^{4}.}"></span></dd></dl> <p>and so </p> <dl><dd><span class="mwe-math-element"><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} =(c(n-1)-1)^{2}\sigma ^{4}+2c^{2}(n-1)\sigma ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>MSE</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {MSE} =(c(n-1)-1)^{2}\sigma ^{4}+2c^{2}(n-1)\sigma ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85570d19eebea0018b19f2169fa1b05b24a12776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.225ex; height:3.176ex;" alt="{\displaystyle \operatorname {MSE} =(c(n-1)-1)^{2}\sigma ^{4}+2c^{2}(n-1)\sigma ^{4}}"></span></dd></dl> <p>With a little algebra it can be confirmed that it is <i>c</i> = 1/(<i>n</i> + 1) which minimises this combined loss function, rather than <i>c</i> = 1/(<i>n</i> − 1) which minimises just the square of the bias. </p><p>More generally it is only in restricted classes of problems that there will be an estimator that minimises the MSE independently of the parameter values. </p><p>However it is very common that there may be perceived to be a <i><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">bias–variance tradeoff</a></i>, such that a small increase in bias can be traded for a larger decrease in variance, resulting in a more desirable estimator overall. </p> <div class="mw-heading mw-heading2"><h2 id="Bayesian_view">Bayesian view</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=11" title="Edit section: Bayesian view"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most bayesians are rather unconcerned about unbiasedness (at least in the formal sampling-theory sense above) of their estimates. For example, Gelman and coauthors (1995) write: "From a Bayesian perspective, the principle of unbiasedness is reasonable in the limit of large samples, but otherwise it is potentially misleading."<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Fundamentally, the difference between the <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian approach</a> and the sampling-theory approach above is that in the sampling-theory approach the parameter is taken as fixed, and then probability distributions of a statistic are considered, based on the predicted sampling distribution of the data. For a Bayesian, however, it is the <i>data</i> which are known, and fixed, and it is the unknown parameter for which an attempt is made to construct a probability distribution, using <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\theta \mid D,I)\propto p(\theta \mid I)p(D\mid \theta ,I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>D</mi> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>∣</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo>∣</mo> <mi>θ</mi> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\theta \mid D,I)\propto p(\theta \mid I)p(D\mid \theta ,I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08330ad621e0dbbbccce24f936d3a7b7d413670f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:30.639ex; height:2.843ex;" alt="{\displaystyle p(\theta \mid D,I)\propto p(\theta \mid I)p(D\mid \theta ,I)}"></span></dd></dl> <p>Here the second term, the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood</a> of the data given the unknown parameter value θ, depends just on the data obtained and the modelling of the data generation process. However a Bayesian calculation also includes the first term, the <a href="/wiki/Prior_probability" title="Prior probability">prior probability</a> for θ, which takes account of everything the analyst may know or suspect about θ <i>before</i> the data comes in. This information plays no part in the sampling-theory approach; indeed any attempt to include it would be considered "bias" away from what was pointed to purely by the data. To the extent that Bayesian calculations include prior information, it is therefore essentially inevitable that their results will not be "unbiased" in sampling theory terms. </p><p>But the results of a Bayesian approach can differ from the sampling theory approach even if the Bayesian tries to adopt an "uninformative" prior. </p><p>For example, consider again the estimation of an unknown population variance σ<sup>2</sup> of a Normal distribution with unknown mean, where it is desired to optimise <i>c</i> in the expected loss function </p> <dl><dd><span class="mwe-math-element"><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 {ExpectedLoss} =\operatorname {E} \left[\left(cnS^{2}-\sigma ^{2}\right)^{2}\right]=\operatorname {E} \left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ExpectedLoss</mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ExpectedLoss} =\operatorname {E} \left[\left(cnS^{2}-\sigma ^{2}\right)^{2}\right]=\operatorname {E} \left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ef172000da21a996abc01468612bdce011ae3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.607ex; height:6.176ex;" alt="{\displaystyle \operatorname {ExpectedLoss} =\operatorname {E} \left[\left(cnS^{2}-\sigma ^{2}\right)^{2}\right]=\operatorname {E} \left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"></span></dd></dl> <p>A standard choice of uninformative prior for this problem is the <a href="/wiki/Jeffreys_prior#Gaussian_distribution_with_standard_deviation_parameter" title="Jeffreys prior">Jeffreys prior</a>, <span class="mwe-math-element"><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 {p(\sigma ^{2})\;\propto \;1/\sigma ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>∝</mo> <mspace width="thickmathspace" /> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {p(\sigma ^{2})\;\propto \;1/\sigma ^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8de73fa8fcaf0ef3c335e5ebe02df5ace71bf84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.063ex; width:9.927ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {p(\sigma ^{2})\;\propto \;1/\sigma ^{2}}}"></span>, which is equivalent to adopting a rescaling-invariant flat prior for <b>ln(σ<sup>2</sup>)</b>. </p><p>One consequence of adopting this prior is that <i>S</i><sup>2</sup>/σ<sup>2</sup> remains a <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">pivotal quantity</a>, i.e. the probability distribution of <i>S</i><sup>2</sup>/σ<sup>2</sup> depends only on <i>S</i><sup>2</sup>/σ<sup>2</sup>, independent of the value of <i>S</i><sup>2</sup> or σ<sup>2</sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid S^{2}\right)=p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid \sigma ^{2}\right)=g\left({\tfrac {S^{2}}{\sigma ^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>∣</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>∣</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid S^{2}\right)=p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid \sigma ^{2}\right)=g\left({\tfrac {S^{2}}{\sigma ^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc72ddccad2072065a5bf79b0eabaf0ce1b544a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:36.295ex; height:4.843ex;" alt="{\displaystyle p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid S^{2}\right)=p\left({\tfrac {S^{2}}{\sigma ^{2}}}\mid \sigma ^{2}\right)=g\left({\tfrac {S^{2}}{\sigma ^{2}}}\right)}"></span></dd></dl> <p>However, while </p> <dl><dd><span class="mwe-math-element"><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} _{p(S^{2}\mid \sigma ^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]=\sigma ^{4}\operatorname {E} _{p(S^{2}\mid \sigma ^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{p(S^{2}\mid \sigma ^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]=\sigma ^{4}\operatorname {E} _{p(S^{2}\mid \sigma ^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb4ffe91fbc418542932416d2935027d677aa01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.238ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} _{p(S^{2}\mid \sigma ^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]=\sigma ^{4}\operatorname {E} _{p(S^{2}\mid \sigma ^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"></span></dd></dl> <p>in contrast </p> <dl><dd><span class="mwe-math-element"><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} _{p(\sigma ^{2}\mid S^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]\neq \sigma ^{4}\operatorname {E} _{p(\sigma ^{2}\mid S^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>≠</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{p(\sigma ^{2}\mid S^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]\neq \sigma ^{4}\operatorname {E} _{p(\sigma ^{2}\mid S^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3916789974195073e695ae4ec06d82b58dd71bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.238ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} _{p(\sigma ^{2}\mid S^{2})}\left[\sigma ^{4}\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]\neq \sigma ^{4}\operatorname {E} _{p(\sigma ^{2}\mid S^{2})}\left[\left(cn{\tfrac {S^{2}}{\sigma ^{2}}}-1\right)^{2}\right]}"></span></dd></dl> <p>— when the expectation is taken over the probability distribution of σ<sup>2</sup> given <i>S</i><sup>2</sup>, as it is in the Bayesian case, rather than <i>S</i><sup>2</sup> given σ<sup>2</sup>, one can no longer take σ<sup>4</sup> as a constant and factor it out. The consequence of this is that, compared to the sampling-theory calculation, the Bayesian calculation puts more weight on larger values of σ<sup>2</sup>, properly taking into account (as the sampling-theory calculation cannot) that under this squared-loss function the consequence of underestimating large values of σ<sup>2</sup> is more costly in squared-loss terms than that of overestimating small values of σ<sup>2</sup>. </p><p>The worked-out Bayesian calculation gives a <a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">scaled inverse chi-squared distribution</a> with <i>n</i> − 1 degrees of freedom for the posterior probability distribution of σ<sup>2</sup>. The expected loss is minimised when <i>cnS</i><sup>2</sup> = <σ<sup>2</sup>>; this occurs when <i>c</i> = 1/(<i>n</i> − 3). </p><p>Even with an uninformative prior, therefore, a Bayesian calculation may not give the same expected-loss minimising result as the corresponding sampling-theory calculation. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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title="Efficient estimator">Efficient estimator</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Expected_loss" title="Expected loss">Expected loss</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Loss_function" title="Loss function">Loss function</a></li> <li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Minimum-variance unbiased estimator</a></li> <li><a href="/wiki/Omitted-variable_bias" title="Omitted-variable bias">Omitted-variable bias</a></li> <li><a href="/wiki/Optimism_bias" title="Optimism bias">Optimism bias</a></li> <li><a href="/wiki/Ratio_estimator" title="Ratio estimator">Ratio estimator</a></li> <li><a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">Statistical decision theory</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 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.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/681638/for-the-binomial-distribution-why-does-no-unbiased-estimator-exist-for-1-p">"For the binomial distribution, why does no unbiased estimator exist for $1/p$?"</a>. <i>Mathematics Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-12-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematics+Stack+Exchange&rft.atitle=For+the+binomial+distribution%2C+why+does+no+unbiased+estimator+exist+for+%241%2Fp%24%3F&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F681638%2Ffor-the-binomial-distribution-why-does-no-unbiased-estimator-exist-for-1-p&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKozdron2016" class="citation web cs1">Kozdron, Michael (March 2016). <a rel="nofollow" class="external text" href="http://stat.math.uregina.ca/~kozdron/Teaching/Regina/252Winter16/Handouts/ch3.pdf">"Evaluating the Goodness of an Estimator: Bias, Mean-Square Error, Relative Efficiency (Chapter 3)"</a> <span class="cs1-format">(PDF)</span>. <i>stat.math.uregina.ca</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=stat.math.uregina.ca&rft.atitle=Evaluating+the+Goodness+of+an+Estimator%3A+Bias%2C+Mean-Square+Error%2C+Relative+Efficiency+%28Chapter+3%29&rft.date=2016-03&rft.aulast=Kozdron&rft.aufirst=Michael&rft_id=http%3A%2F%2Fstat.math.uregina.ca%2F~kozdron%2FTeaching%2FRegina%2F252Winter16%2FHandouts%2Fch3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor2019" class="citation web cs1">Taylor, Courtney (January 13, 2019). <a rel="nofollow" class="external text" href="https://www.thoughtco.com/what-is-an-unbiased-estimator-3126502">"Unbiased and Biased Estimators"</a>. <i>ThoughtCo</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ThoughtCo&rft.atitle=Unbiased+and+Biased+Estimators&rft.date=2019-01-13&rft.aulast=Taylor&rft.aufirst=Courtney&rft_id=https%3A%2F%2Fwww.thoughtco.com%2Fwhat-is-an-unbiased-estimator-3126502&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekking2005" class="citation book cs1">Dekking, Michel, ed. (2005). <i>A modern introduction to probability and statistics: understanding why and how</i>. Springer texts in statistics. London [Heidelberg]: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-896-1" title="Special:BookSources/978-1-85233-896-1"><bdi>978-1-85233-896-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+modern+introduction+to+probability+and+statistics%3A+understanding+why+and+how&rft.place=London+%5BHeidelberg%5D&rft.series=Springer+texts+in+statistics&rft.pub=Springer&rft.date=2005&rft.isbn=978-1-85233-896-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-JohnsonWichern2007-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-JohnsonWichern2007_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Arnold_JohnsonDean_W._Wichern2007" class="citation book cs1">Richard Arnold Johnson; Dean W. Wichern (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gFWcQgAACAAJ"><i>Applied Multivariate Statistical Analysis</i></a>. Pearson Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-187715-3" title="Special:BookSources/978-0-13-187715-3"><bdi>978-0-13-187715-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">10 August</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Multivariate+Statistical+Analysis&rft.pub=Pearson+Prentice+Hall&rft.date=2007&rft.isbn=978-0-13-187715-3&rft.au=Richard+Arnold+Johnson&rft.au=Dean+W.+Wichern&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgFWcQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRomanoSiegel1986" class="citation book cs1">Romano, J. P.; Siegel, A. F. (1986). <a href="/wiki/Counterexamples_in_Probability_and_Statistics" title="Counterexamples in Probability and Statistics"><i>Counterexamples in Probability and Statistics</i></a>. Monterey, California, USA: Wadsworth & Brooks / Cole. p. 168.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counterexamples+in+Probability+and+Statistics&rft.place=Monterey%2C+California%2C+USA&rft.pages=168&rft.pub=Wadsworth+%26+Brooks+%2F+Cole&rft.date=1986&rft.aulast=Romano&rft.aufirst=J.+P.&rft.au=Siegel%2C+A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy2003" class="citation journal cs1">Hardy, M. (1 March 2003). "An Illuminating Counterexample". <i>American Mathematical Monthly</i>. <b>110</b> (3): 234–238. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0206006">math/0206006</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3647938">10.2307/3647938</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3647938">3647938</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=An+Illuminating+Counterexample&rft.volume=110&rft.issue=3&rft.pages=234-238&rft.date=2003-03-01&rft_id=info%3Aarxiv%2Fmath%2F0206006&rft.issn=0002-9890&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3647938%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3647938&rft.aulast=Hardy&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Brown (1947), page 583</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFLehmann1951">Lehmann 1951</a>; <a href="#CITEREFBirnbaum1961">Birnbaum 1961</a>; <a href="#CITEREFVan_der_Vaart1961">Van der Vaart 1961</a>; <a href="#CITEREFPfanzagl1994">Pfanzagl 1994</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfanzagl1979" class="citation journal cs1">Pfanzagl, Johann (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344563">"On optimal median unbiased estimators in the presence of nuisance parameters"</a>. <i>The Annals of Statistics</i>. <b>7</b> (1): 187–193. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344563">10.1214/aos/1176344563</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=On+optimal+median+unbiased+estimators+in+the+presence+of+nuisance+parameters&rft.volume=7&rft.issue=1&rft.pages=187-193&rft.date=1979&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176344563&rft.aulast=Pfanzagl&rft.aufirst=Johann&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176344563&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-BrownEtAl-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-BrownEtAl_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BrownEtAl_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrownCohenStrawderman1976" class="citation journal cs1">Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343543">"A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications"</a>. <i>Ann. Statist</i>. <b>4</b> (4): 712–722. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343543">10.1214/aos/1176343543</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Statist.&rft.atitle=A+Complete+Class+Theorem+for+Strict+Monotone+Likelihood+Ratio+With+Applications&rft.volume=4&rft.issue=4&rft.pages=712-722&rft.date=1976&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176343543&rft.aulast=Brown&rft.aufirst=L.+D.&rft.au=Cohen%2C+Arthur&rft.au=Strawderman%2C+W.+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176343543&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-Dodge-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dodge_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dodge_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Dodge_12-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDodge1987" class="citation book cs1">Dodge, Yadolah, ed. (1987). <i>Statistical Data Analysis Based on the L<sub>1</sub>-Norm and Related Methods</i>. Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987. Amsterdam: North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-70273-3" title="Special:BookSources/0-444-70273-3"><bdi>0-444-70273-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Data+Analysis+Based+on+the+L%3Csub%3E1%3C%2Fsub%3E-Norm+and+Related+Methods&rft.place=Amsterdam&rft.series=Papers+from+the+First+International+Conference+held+at+Neuch%C3%A2tel%2C+August+31%E2%80%93September+4%2C+1987&rft.pub=North-Holland&rft.date=1987&rft.isbn=0-444-70273-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaynes2007" class="citation book cs1">Jaynes, E. T. (2007). <i>Probability Theory : The Logic of Science</i>. Cambridge: Cambridge Univ. Press. p. 172. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-59271-0" title="Special:BookSources/978-0-521-59271-0"><bdi>978-0-521-59271-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+Theory+%3A+The+Logic+of+Science&rft.place=Cambridge&rft.pages=172&rft.pub=Cambridge+Univ.+Press&rft.date=2007&rft.isbn=978-0-521-59271-0&rft.aulast=Jaynes&rft.aufirst=E.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlebanovRachevFabozzi2009" class="citation book cs1">Klebanov, Lev B.; Rachev, Svetlozar T.; Fabozzi, Frank J. (2009). "Loss Functions and the Theory of Unbiased Estimation". <i>Robust and Non-Robust Models in Statistics</i>. New York: Nova Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60741-768-2" title="Special:BookSources/978-1-60741-768-2"><bdi>978-1-60741-768-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Loss+Functions+and+the+Theory+of+Unbiased+Estimation&rft.btitle=Robust+and+Non-Robust+Models+in+Statistics&rft.place=New+York&rft.pub=Nova+Scientific&rft.date=2009&rft.isbn=978-1-60741-768-2&rft.aulast=Klebanov&rft.aufirst=Lev+B.&rft.au=Rachev%2C+Svetlozar+T.&rft.au=Fabozzi%2C+Frank+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> <li id="cite_note-taboga-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-taboga_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaboga2010" class="citation web cs1">Taboga, Marco (2010). <a rel="nofollow" class="external text" href="http://www.statlect.com/glossary/mean_squared_error.htm">"Lectures on probability theory and mathematical statistics"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lectures+on+probability+theory+and+mathematical+statistics&rft.date=2010&rft.aulast=Taboga&rft.aufirst=Marco&rft_id=http%3A%2F%2Fwww.statlect.com%2Fglossary%2Fmean_squared_error.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeGroot1986" class="citation book cs1">DeGroot, Morris H. (1986). <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/probabilitystati00degr_679"><i>Probability and Statistics</i></a></span> (2nd ed.). Addison-Wesley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilitystati00degr_679/page/n423">414</a>–5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-11366-X" title="Special:BookSources/0-201-11366-X"><bdi>0-201-11366-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Statistics&rft.pages=414-5&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1986&rft.isbn=0-201-11366-X&rft.aulast=DeGroot&rft.aufirst=Morris+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprobabilitystati00degr_679&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span> But compare it with, for example, the discussion in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasellaBerger2001" class="citation book cs1">Casella; Berger (2001). <i>Statistical Inference</i> (2nd ed.). Duxbury. p. 332. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-24312-6" title="Special:BookSources/0-534-24312-6"><bdi>0-534-24312-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Inference&rft.pages=332&rft.edition=2nd&rft.pub=Duxbury&rft.date=2001&rft.isbn=0-534-24312-6&rft.au=Casella&rft.au=Berger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGelmanCarlinSternRubin1995" class="citation book cs1">Gelman, A.; et al. (1995). <i>Bayesian Data Analysis</i>. Chapman and Hall. p. 108. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-03991-5" title="Special:BookSources/0-412-03991-5"><bdi>0-412-03991-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Data+Analysis&rft.pages=108&rft.pub=Chapman+and+Hall&rft.date=1995&rft.isbn=0-412-03991-5&rft.aulast=Gelman&rft.aufirst=A.&rft.au=Carlin%2C+John+S.&rft.au=Stern%2C+Hal+S.&rft.au=Rubin%2C+Donald+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110310043642/http://www.universityofcalifornia.edu/senate/inmemoriam/georgewbrown.htm">Brown, George W.</a> "On Small-Sample Estimation." <i>The Annals of Mathematical Statistics</i>, vol. 18, no. 4 (Dec., 1947), pp. 582–585. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236236">2236236</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehmann1951" class="citation journal cs1"><a href="/wiki/Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, E. L.</a> (December 1951). "A General Concept of Unbiasedness". <i>The Annals of Mathematical Statistics</i>. <b>22</b> (4): 587–592. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236928">2236928</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.atitle=A+General+Concept+of+Unbiasedness&rft.volume=22&rft.issue=4&rft.pages=587-592&rft.date=1951-12&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236928%23id-name%3DJSTOR&rft.aulast=Lehmann&rft.aufirst=E.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirnbaum1961" class="citation journal cs1"><a href="/wiki/Allan_Birnbaum" title="Allan Birnbaum">Birnbaum, Allan</a> (March 1961). "A Unified Theory of Estimation, I". <i>The Annals of Mathematical Statistics</i>. <b>32</b> (1): 112–135.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.atitle=A+Unified+Theory+of+Estimation%2C+I&rft.volume=32&rft.issue=1&rft.pages=112-135&rft.date=1961-03&rft.aulast=Birnbaum&rft.aufirst=Allan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_der_Vaart1961" class="citation journal cs1">Van der Vaart, H. R. (June 1961). <a rel="nofollow" class="external text" href="https://projecteuclid.org/download/pdf_1/euclid.aoms/1177705051">"Some Extensions of the Idea of Bias"</a>. <i>The Annals of Mathematical Statistics</i>. <b>32</b> (2): 436–447.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.atitle=Some+Extensions+of+the+Idea+of+Bias&rft.volume=32&rft.issue=2&rft.pages=436-447&rft.date=1961-06&rft.aulast=Van+der+Vaart&rft.aufirst=H.+R.&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.aoms%2F1177705051&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfanzagl1994" class="citation book cs1">Pfanzagl, Johann (1994). <i>Parametric Statistical Theory</i>. Walter de Gruyter.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parametric+Statistical+Theory&rft.pub=Walter+de+Gruyter&rft.date=1994&rft.aulast=Pfanzagl&rft.aufirst=Johann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStuartOrdArnold2010" class="citation book cs1">Stuart, Alan; Ord, Keith; Arnold, Steven [F.] (2010). <i>Classical Inference and the Linear Model</i>. Kendall's Advanced Theory of Statistics. Vol. 2A. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-4706-8924-0" title="Special:BookSources/978-0-4706-8924-0"><bdi>978-0-4706-8924-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Inference+and+the+Linear+Model&rft.series=Kendall%27s+Advanced+Theory+of+Statistics&rft.pub=Wiley&rft.date=2010&rft.isbn=978-0-4706-8924-0&rft.aulast=Stuart&rft.aufirst=Alan&rft.au=Ord%2C+Keith&rft.au=Arnold%2C+Steven+%5BF.%5D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoinovNikulin1993" class="citation book cs1 cs1-prop-long-vol">Voinov, Vassily [G.]; Nikulin, Mikhail [S.] (1993). <i>Unbiased estimators and their applications</i>. Vol. 1: Univariate case. Dordrect: Kluwer Academic Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2382-3" title="Special:BookSources/0-7923-2382-3"><bdi>0-7923-2382-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unbiased+estimators+and+their+applications&rft.place=Dordrect&rft.pub=Kluwer+Academic+Publishers&rft.date=1993&rft.isbn=0-7923-2382-3&rft.aulast=Voinov&rft.aufirst=Vassily+%5BG.%5D&rft.au=Nikulin%2C+Mikhail+%5BS.%5D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoinovNikulin1996" class="citation book cs1 cs1-prop-long-vol">Voinov, Vassily [G.]; Nikulin, Mikhail [S.] (1996). <i>Unbiased estimators and their applications</i>. Vol. 2: Multivariate case. Dordrect: Kluwer Academic Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-3939-8" title="Special:BookSources/0-7923-3939-8"><bdi>0-7923-3939-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unbiased+estimators+and+their+applications&rft.place=Dordrect&rft.pub=Kluwer+Academic+Publishers&rft.date=1996&rft.isbn=0-7923-3939-8&rft.aulast=Voinov&rft.aufirst=Vassily+%5BG.%5D&rft.au=Nikulin%2C+Mikhail+%5BS.%5D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlebanovRachevFabozzi2009" class="citation book cs1">Klebanov, Lev [B.]; Rachev, Svetlozar [T.]; Fabozzi, Frank [J.] (2009). <i>Robust and Non-Robust Models in Statistics</i>. New York: Nova Scientific Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60741-768-2" title="Special:BookSources/978-1-60741-768-2"><bdi>978-1-60741-768-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robust+and+Non-Robust+Models+in+Statistics&rft.place=New+York&rft.pub=Nova+Scientific+Publishers&rft.date=2009&rft.isbn=978-1-60741-768-2&rft.aulast=Klebanov&rft.aufirst=Lev+%5BB.%5D&rft.au=Rachev%2C+Svetlozar+%5BT.%5D&rft.au=Fabozzi%2C+Frank+%5BJ.%5D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bias_of_an_estimator&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Unbiased_estimator">"Unbiased estimator"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Unbiased+estimator&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DUnbiased_estimator&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias+of+an+estimator" class="Z3988"></span></li></ul> <div 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title="Outline of statistics">Outline</a></li> <li><a href="/wiki/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_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/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="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/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="/wiki/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="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/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_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/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="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/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_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a class="mw-selflink selflink">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/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="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/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="/wiki/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="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/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="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/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="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/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="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/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="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/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="/wiki/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="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/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_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/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="/wiki/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="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/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="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/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="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/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="/wiki/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="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/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_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/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="/wiki/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="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/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="/wiki/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="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/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="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/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="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li> <li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Biases" title="Template:Biases"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Biases" title="Template talk:Biases"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Biases" title="Special:EditPage/Template:Biases"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Biases" style="font-size:114%;margin:0 4em"><a href="/wiki/Bias" title="Bias">Biases</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Cognitive_bias" title="Cognitive bias">Cognitive biases</a></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acquiescence_bias" title="Acquiescence bias">Acquiescence</a></li> <li><a href="/wiki/Ambiguity_effect" title="Ambiguity effect">Ambiguity</a></li> <li><a href="/wiki/Affinity_bias" title="Affinity bias">Affinity</a></li> <li><a href="/wiki/Anchoring_(cognitive_bias)" class="mw-redirect" title="Anchoring (cognitive bias)">Anchoring</a></li> <li><a href="/wiki/Attentional_bias" title="Attentional bias">Attentional</a></li> <li><a href="/wiki/Attribution_bias" title="Attribution bias">Attribution</a> <ul><li><a href="/wiki/Actor%E2%80%93observer_asymmetry" title="Actor–observer asymmetry">Actor–observer</a></li> <li><a href="/wiki/Fundamental_attribution_error" title="Fundamental attribution error">Correspondence</a></li></ul></li> <li><a href="/wiki/Authority_bias" title="Authority bias">Authority</a></li> <li><a href="/wiki/Automation_bias" title="Automation bias">Automation</a></li> <li><a href="/wiki/Availability_heuristic" title="Availability heuristic">Availability</a> <ul><li><a href="/wiki/Mean_world_syndrome" title="Mean world syndrome">Mean world</a></li></ul></li> <li><a href="/wiki/Belief_bias" title="Belief bias">Belief</a></li> <li><a href="/wiki/Bias_blind_spot" title="Bias blind spot">Blind spot</a></li> <li><a href="/wiki/Choice-supportive_bias" title="Choice-supportive bias">Choice-supportive</a></li> <li><a href="/wiki/Escalation_of_commitment" title="Escalation of commitment">Commitment</a></li> <li><a href="/wiki/Confirmation_bias" title="Confirmation bias">Confirmation</a> <ul><li><a href="/wiki/Selective_perception" title="Selective perception">Selective perception</a></li></ul></li> <li><a href="/wiki/Compassion_fade" title="Compassion fade">Compassion fade</a></li> <li><a href="/wiki/Congruence_bias" title="Congruence bias">Congruence</a></li> <li><a href="/wiki/Cultural_bias" title="Cultural bias">Cultural</a></li> <li><a href="/wiki/Declinism" title="Declinism">Declinism</a></li> <li><a href="/wiki/Distinction_bias" title="Distinction bias">Distinction</a></li> <li><a href="/wiki/Dunning%E2%80%93Kruger_effect" title="Dunning–Kruger effect">Dunning–Kruger</a></li> <li><a href="/wiki/Egocentric_bias" title="Egocentric bias">Egocentric</a> <ul><li><a href="/wiki/Curse_of_knowledge" title="Curse of knowledge">Curse of knowledge</a></li></ul></li> <li><a href="/wiki/Emotional_bias" title="Emotional bias">Emotional</a></li> <li><a href="/wiki/Extrinsic_incentives_bias" title="Extrinsic incentives bias">Extrinsic incentives</a></li> <li><a href="/wiki/Fading_affect_bias" title="Fading affect bias">Fading affect</a></li> <li><a href="/wiki/Framing_effect_(psychology)" title="Framing effect (psychology)">Framing</a></li> <li><a href="/wiki/Frequency_illusion" title="Frequency illusion">Frequency</a></li> <li><a href="/wiki/Frog_pond_effect" title="Frog pond effect">Frog pond effect</a></li> <li><a href="/wiki/Halo_effect" title="Halo effect">Halo effect</a></li> <li><a href="/wiki/Hindsight_bias" title="Hindsight bias">Hindsight</a></li> <li><a href="/wiki/Horn_effect" title="Horn effect">Horn effect</a></li> <li><a href="/wiki/Hostile_attribution_bias" title="Hostile attribution bias">Hostile attribution</a></li> <li><a href="/wiki/Impact_bias" title="Impact bias">Impact</a></li> <li><a href="/wiki/Implicit_stereotype" title="Implicit stereotype">Implicit</a></li> <li><a href="/wiki/In-group_favoritism" title="In-group favoritism">In-group</a></li> <li><a href="/wiki/Intentionality_bias" title="Intentionality bias">Intentionality</a></li> <li><a href="/wiki/Illusion_of_transparency" title="Illusion of transparency">Illusion of transparency</a></li> <li><a href="/wiki/Mean_world_syndrome" title="Mean world syndrome">Mean world syndrome</a></li> <li><a href="/wiki/Mere-exposure_effect" title="Mere-exposure effect">Mere-exposure effect</a></li> <li><a href="/wiki/Narrative_bias" title="Narrative bias">Narrative</a></li> <li><a href="/wiki/Negativity_bias" title="Negativity bias">Negativity</a></li> <li><a href="/wiki/Normalcy_bias" title="Normalcy bias">Normalcy</a></li> <li><a href="/wiki/Omission_bias" title="Omission bias">Omission</a></li> <li><a href="/wiki/Optimism_bias" title="Optimism bias">Optimism</a></li> <li><a href="/wiki/Out-group_homogeneity" title="Out-group homogeneity">Out-group homogeneity</a></li> <li><a href="/wiki/Outcome_bias" title="Outcome bias">Outcome</a></li> <li><a href="/wiki/Overton_window" title="Overton window">Overton window</a></li> <li><a href="/wiki/Precision_bias" title="Precision bias">Precision</a></li> <li><a href="/wiki/Present_bias" title="Present bias">Present</a></li> <li><a href="/wiki/Pro-innovation_bias" title="Pro-innovation bias">Pro-innovation</a></li> <li><a href="/wiki/Proximity_bias" title="Proximity bias">Proximity</a></li> <li><a href="/wiki/Response_bias" title="Response bias">Response</a></li> <li><a href="/wiki/Restraint_bias" title="Restraint bias">Restraint</a></li> <li><a href="/wiki/Self-serving_bias" title="Self-serving bias">Self-serving</a></li> <li><a href="/wiki/Social_comparison_bias" title="Social comparison bias">Social comparison</a></li> <li><a href="/wiki/Social_influence_bias" title="Social influence bias">Social influence bias</a></li> <li><a href="/wiki/Spotlight_effect" title="Spotlight effect">Spotlight</a></li> <li><a href="/wiki/Status_quo_bias" title="Status quo bias">Status quo</a></li> <li><a href="/wiki/Attribute_substitution" title="Attribute substitution">Substitution</a></li> <li><a href="/wiki/Time-saving_bias" title="Time-saving bias">Time-saving</a></li> <li><a href="/wiki/Trait_ascription_bias" title="Trait ascription bias">Trait ascription</a></li> <li><a href="/wiki/Turkey_illusion" title="Turkey illusion">Turkey illusion</a></li> <li><a href="/wiki/Von_Restorff_effect" title="Von Restorff effect">von Restorff effect</a></li> <li><a href="/wiki/Zero-risk_bias" title="Zero-risk bias">Zero-risk</a></li> <li><a href="/wiki/Cognitive_bias_in_animals" title="Cognitive bias in animals">In animals</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bias_(statistics)" title="Bias (statistics)">Statistical biases</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Estimator</a></li> <li><a href="/wiki/Forecast_bias" title="Forecast bias">Forecast</a></li> <li><a href="/wiki/Healthy_user_bias" title="Healthy user bias">Healthy user</a></li> <li><a href="/wiki/Information_bias_(epidemiology)" title="Information bias (epidemiology)">Information</a> <ul><li><a href="/wiki/Information_bias_(psychology)" title="Information bias (psychology)">Psychological</a></li></ul></li> <li><a href="/wiki/Lead_time_bias" title="Lead time bias">Lead time</a></li> <li><a href="/wiki/Length_time_bias" title="Length time bias">Length time</a></li> <li><a href="/wiki/Participation_bias" title="Participation bias">Non-response</a></li> <li><a href="/wiki/Observer_bias" title="Observer bias">Observer</a></li> <li><a href="/wiki/Omitted-variable_bias" title="Omitted-variable bias">Omitted-variable</a></li> <li><a href="/wiki/Participation_bias" title="Participation bias">Participation</a></li> <li><a href="/wiki/Recall_bias" title="Recall bias">Recall</a></li> <li><a href="/wiki/Sampling_bias" title="Sampling bias">Sampling</a></li> <li><a href="/wiki/Selection_bias" title="Selection bias">Selection</a></li> <li><a href="/wiki/Self-selection_bias" title="Self-selection bias">Self-selection</a></li> <li><a href="/wiki/Social-desirability_bias" title="Social-desirability bias">Social desirability</a></li> <li><a href="/wiki/Spectrum_bias" title="Spectrum bias">Spectrum</a></li> <li><a href="/wiki/Survivorship_bias" title="Survivorship bias">Survivorship</a></li> <li><a href="/wiki/Systematic_error" class="mw-redirect" title="Systematic error">Systematic error</a></li> <li><a href="/wiki/Systemic_bias" title="Systemic bias">Systemic</a></li> <li><a href="/wiki/Verification_bias" title="Verification bias">Verification</a></li> <li><a href="/wiki/Wet_bias" title="Wet bias">Wet</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other biases</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Academic_bias" title="Academic bias">Academic</a></li> <li><a href="/wiki/Basking_in_reflected_glory" title="Basking in reflected glory">Basking in reflected glory</a></li> <li><i><a href="/wiki/D%C3%A9formation_professionnelle" title="Déformation professionnelle">Déformation professionnelle</a></i></li> <li><a href="/wiki/Funding_bias" title="Funding bias">Funding</a></li> <li><a href="/wiki/FUTON_bias" class="mw-redirect" title="FUTON bias">FUTON</a></li> <li><a href="/wiki/Inductive_bias" title="Inductive bias">Inductive</a></li> <li><a href="/wiki/Infrastructure_bias" title="Infrastructure bias">Infrastructure</a></li> <li><a href="/wiki/Inherent_bias" title="Inherent bias">Inherent</a></li> <li><a href="/wiki/Bias_in_education" class="mw-redirect" title="Bias in education">In education</a></li> <li><a href="/wiki/Liking_gap" title="Liking gap">Liking gap</a></li> <li><a href="/wiki/Media_bias" title="Media bias">Media</a> <ul><li><a href="/wiki/False_balance" title="False balance">False balance</a></li> <li><a href="/wiki/United_States_news_media_and_the_Vietnam_War" title="United States news media and the Vietnam War">Vietnam War</a></li> <li><a href="/wiki/Media_of_Norway" class="mw-redirect" title="Media of Norway">Norway</a></li> <li><a href="/wiki/Media_bias_in_South_Asia" title="Media bias in South Asia">South Asia</a></li> <li><a href="/wiki/Media_of_Sweden" class="mw-redirect" title="Media of Sweden">Sweden</a></li> <li><a href="/wiki/Media_bias_in_the_United_States" title="Media bias in the United States">United States</a></li> <li><a href="/wiki/Media_coverage_of_the_Arab%E2%80%93Israeli_conflict" class="mw-redirect" title="Media coverage of the Arab–Israeli conflict">Arab–Israeli conflict</a></li> <li><a href="/wiki/Media_portrayal_of_the_Ukrainian_crisis" class="mw-redirect" title="Media portrayal of the Ukrainian crisis">Ukraine</a></li></ul></li> <li><a href="/wiki/Net_bias" title="Net bias">Net</a></li> <li><a href="/wiki/Political_bias" title="Political bias">Political bias</a></li> <li><a href="/wiki/Publication_bias" title="Publication bias">Publication</a></li> <li><a href="/wiki/Reporting_bias" title="Reporting bias">Reporting</a></li> <li><a href="/wiki/White_hat_bias" title="White hat bias">White hat</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bias reduction</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cognitive_bias_mitigation" title="Cognitive bias mitigation">Cognitive bias mitigation</a></li> <li><a href="/wiki/Debiasing" title="Debiasing">Debiasing</a></li> <li><a href="/wiki/Heuristic_(psychology)" title="Heuristic (psychology)">Heuristics in judgment and decision-making</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li>Lists: <a href="/wiki/List_of_cognitive_biases" title="List of cognitive biases">General</a></li> <li><a href="/wiki/List_of_cognitive_biases#Memory_biases" title="List of cognitive biases">Memory</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" 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Bias of an estimator - Wikipedia