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Stein's example

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In <a href="Decision_theory" title="Decision theory">decision theory</a> and <a href="Estimation_theory" title="Estimation theory">estimation theory</a>, Stein's example (also known as Stein's phenomenon or Stein's paradox) is the observation that when three or more parameters are estimated simultaneously, there exist combined <a href="Estimator" title="Estimator">estimators</a> more accurate on average (that is, having lower expected <a href="Mean_squared_error" title="Mean squared error">mean squared error</a>) than any method that handles the parameters separately. It is named after <a href="Charles_Stein_(statistician)" class="mw-redirect" title="Charles Stein (statistician)">Charles Stein</a> of <a href="Stanford_University" title="Stanford University">Stanford University</a>, who discovered the phenomenon in 1955.<a href="#cite_note-1">[1]</a>

An intuitive explanation is that optimizing for the mean-squared error of a combined estimator is not the same as optimizing for the errors of separate estimators of the individual parameters. In practical terms, if the combined error is in fact of interest, then a combined estimator should be used, even if the underlying parameters are independent. If one is instead interested in estimating an individual parameter, then using a combined estimator does not help and is in fact worse.

<div class="mw-heading mw-heading2"><h2 id="Formal_statement">Formal statement</h2></div>

The following is the simplest form of the paradox, the special case in which the number of observations is equal to the number of parameters to be estimated. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> be a vector consisting of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

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</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy"> unknown parameters. To estimate these parameters, a single measurement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" 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}}" loading="lazy"> is performed for each parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}">

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

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</math><img src="./302b19204ed378e99ff4575341a67eebdbe5a555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}" loading="lazy">, resulting in a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">. Suppose the measurements are known to be <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>, <a href="Normal_distribution" title="Normal distribution">Gaussian</a> <a href="Random_variable" title="Random variable">random variables</a>, with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> and variance 1, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} \sim {\mathcal {N}}({\boldsymbol {\theta }},\mathbf {I} _{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \sim {\mathcal {N}}({\boldsymbol {\theta }},\mathbf {I} _{n})}</annotation>

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</math><img src="./942a634f9be3a32ffb3c9cfe5f6e1bdec5b7fbc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.775ex; height:3.009ex;" alt="{\displaystyle \mathbf {X} \sim {\mathcal {N}}({\boldsymbol {\theta }},\mathbf {I} _{n})}" loading="lazy">. Thus, each parameter is estimated using a single noisy measurement, and each measurement is equally inaccurate.

Under these conditions, it is intuitive and common to use each measurement as an estimate of its corresponding parameter. This so-called "ordinary" decision rule can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}=\mathbf {X} }">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}=\mathbf {X} }</annotation>

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</math><img src="./9ec5149462fbc90049c9c6a12d9941b087435c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.575ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}=\mathbf {X} }" loading="lazy">, which is the <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimator</a> (MLE). The quality of such an estimator is measured by its <a href="Risk_function" class="mw-redirect" title="Risk function">risk function</a>. A commonly used risk function is the <a href="Mean_squared_error" title="Mean squared error">mean squared error</a>, defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [\|{\boldsymbol {\theta }}-{\hat {\boldsymbol {\theta }}}\|^{2}]}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [\|{\boldsymbol {\theta }}-{\hat {\boldsymbol {\theta }}}\|^{2}]}</annotation>

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</math><img src="./1d8af4d0aa7c1952dd11a9f91057b928a021a09e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.827ex; height:3.343ex;" alt="{\displaystyle \mathbb {E} [\|{\boldsymbol {\theta }}-{\hat {\boldsymbol {\theta }}}\|^{2}]}" loading="lazy">. Surprisingly, it turns out that the "ordinary" decision rule is suboptimal (<a href="Admissible_decision_rule" title="Admissible decision rule">inadmissible</a>) in terms of mean squared error when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

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</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy">. In other words, in the setting discussed here, there exist alternative estimators which always achieve lower mean squared error, no matter what the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> is. For a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> one could obviously define a perfect "estimator" which is always just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">, but this estimator would be bad for other values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">.

The estimators of Stein's paradox are, for a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">, better than the "ordinary" decision rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> but necessarily worse for others. It is only on average that they are better. More accurately, an estimator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}</annotation>

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</math><img src="./591dfd7c44cd6d76ec1166d07a4b8e5db8693ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}" loading="lazy"> is said to <a href="Dominating_decision_rule" title="Dominating decision rule">dominate</a> another estimator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}</annotation>

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</math><img src="./aa0ad5c0da0ee39992006612bd6560d5e0754082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}" loading="lazy"> if, for all values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">, the risk of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}</annotation>

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</math><img src="./591dfd7c44cd6d76ec1166d07a4b8e5db8693ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}_{1}}" loading="lazy"> is lower than, or equal to, the risk of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}</annotation>

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</math><img src="./aa0ad5c0da0ee39992006612bd6560d5e0754082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\theta }}}_{2}}" loading="lazy">, and if the inequality is <a href="Strict" title="Strict">strict</a> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">. An estimator is said to be <a href="Admissible_decision_rule" title="Admissible decision rule">admissible</a> if no other estimator dominates it, otherwise it is inadmissible. Thus, Stein's example can be simply stated as follows: The "ordinary" decision rule of the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk.

Many simple, practical estimators achieve better performance than the "ordinary" decision rule. The best-known example is the <a href="James%E2%80%93Stein_estimator" title="James–Stein estimator">James–Stein estimator</a>, which shrinks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> towards a particular point (such as the origin) by an amount inversely proportional to the distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> from that point. For a sketch of the proof of this result, see <a href="Proof_of_Stein's_example" class="mw-redirect" title="Proof of Stein's example">Proof of Stein's example</a>. An alternative proof is due to <a href="Lawrence_D._Brown" title="Lawrence D. Brown">Larry Brown</a>: he proved that the ordinary estimator for an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">-dimensional multivariate normal mean vector is admissible if and only if the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> -dimensional <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> is recurrent.<a href="#cite_note-2">[2]</a> Since the Brownian motion is not recurrent for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

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</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy">, the MLE is not admissible for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

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</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="An_intuitive_explanation">An intuitive explanation</h2></div>

For any particular value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> the new estimator will improve at least one of the individual mean square errors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [(\theta _{i}-{\hat {\theta }}_{i})^{2}].}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [(\theta _{i}-{\hat {\theta }}_{i})^{2}].}</annotation>

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</math><img src="./cca64a011bd4e53d7ec109a38105d87f7848074b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.241ex; height:3.343ex;" alt="{\displaystyle \mathbb {E} [(\theta _{i}-{\hat {\theta }}_{i})^{2}].}" loading="lazy"> This is not hard − for instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> is between −1 and 1, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1}">

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

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</math><img src="./f759e9b01b4c117d116da9f6d0e635b2247ee502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =1}" loading="lazy">, then an estimator that linearly shrinks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> towards 0 by 0.5 (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sign} (X_{i})\max(|X_{i}|-0.5,0)}">

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</math><img src="./8962eeee09f24bcf1bdcefccec12e4242927be12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.1ex; height:2.843ex;" alt="{\displaystyle \operatorname {sign} (X_{i})\max(|X_{i}|-0.5,0)}" loading="lazy">, soft thresholding with threshold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.5}">

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</math><img src="./c867fe7d5d53ce2c0790852289b794c6ed185f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 0.5}" loading="lazy">) will have a lower mean square error than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> itself. But there are other values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> for which this estimator is worse than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><img src="./9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"> itself. The trick of the Stein estimator, and others that yield the Stein paradox, is that they adjust the shift in such a way that there is always (for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy"> vector) at least one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" 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}}" loading="lazy"> whose mean square error is improved, and its improvement more than compensates for any degradation in mean square error that might occur for another <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\theta }}_{i}}">

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</math><img src="./404e9c4c3f6e80c53ec8231d73323165d8b2bf68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:3.176ex;" alt="{\displaystyle {\hat {\theta }}_{i}}" loading="lazy">. The trouble is that, without knowing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\theta }}}">

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

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</math><img src="./33b025a6bf54ec02e65c871dc3e5897c921419cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\theta }}}" loading="lazy">, you don't know which of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> mean square errors are improved, so you can't use the Stein estimator only for those parameters.

An example of the above setting occurs in <a href="Channel_estimation" class="mw-redirect" title="Channel estimation">channel estimation</a> in telecommunications, for instance, because different factors affect overall channel performance.

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

Stein's example is surprising, since the "ordinary" decision rule is intuitive and commonly used. In fact, numerous methods for estimator construction, including <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a>, <a href="Gauss%E2%80%93Markov_theorem" title="Gauss–Markov theorem">best linear unbiased estimation</a>, <a href="Least_squares" title="Least squares">least squares</a> estimation and optimal <a href="Equivariant_estimation" class="mw-redirect" title="Equivariant estimation">equivariant estimation</a>, all result in the "ordinary" estimator. Yet, as discussed above, this estimator is suboptimal.

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

To demonstrate the unintuitive nature of Stein's example, consider the following real-world example. Suppose we are to estimate three unrelated parameters, such as the US wheat yield for 1993, the number of spectators at the Wimbledon tennis tournament in 2001, and the weight of a randomly chosen candy bar from the supermarket. Suppose we have independent Gaussian measurements of each of these quantities. Stein's example now tells us that we can get a better estimate (on average) for the vector of three parameters by simultaneously using the three unrelated measurements.

At first sight it appears that somehow we get a better estimator for US wheat yield by measuring some other unrelated statistics such as the number of spectators at Wimbledon and the weight of a candy bar. However, we have not obtained a better estimator for US wheat yield by itself, but we have produced an estimator for the vector of the means of all three random variables, which has a reduced total risk. This occurs because the cost of a bad estimate in one component of the vector is compensated by a better estimate in another component. Also, a specific set of the three estimated mean values obtained with the new estimator will not necessarily be better than the ordinary set (the measured values). It is only on average that the new estimator is better.

<div class="mw-heading mw-heading2"><h2 id="Sketched_proof">Sketched proof</h2></div>

The <a href="Risk_function" class="mw-redirect" title="Risk function">risk function</a> of the decision rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {x} )=\mathbf {x} }">

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<annotation encoding="application/x-tex">{\displaystyle d(\mathbf {x} )=\mathbf {x} }</annotation>

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</math><img src="./c4cee9fef3cd65d7b00e990247dc8d9e62e50bb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.946ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {x} )=\mathbf {x} }" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle R(\theta ,d)=\operatorname {E} _{\theta }[|{\boldsymbol {\theta }}-\mathbf {X} |^{2}]}</annotation>

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</math><img src="./a3c037e28fb0862649b853f5b890206932a303ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.406ex; height:3.343ex;" alt="{\displaystyle R(\theta ,d)=\operatorname {E} _{\theta }[|{\boldsymbol {\theta }}-\mathbf {X} |^{2}]}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\int ({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{(-1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx}</annotation>

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</math><img src="./235edc76727c8b6d9f27e6d5a265ad839f916dc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.332ex; height:6.676ex;" alt="{\displaystyle =\int ({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{(-1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx}" loading="lazy"></dd>

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

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</math><img src="./5ab109a6d7399b91da7c630deab21349e715a3f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.495ex; height:1.676ex;" alt="{\displaystyle =n.}" loading="lazy"></dd></dl>

Now consider the decision rule

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<annotation encoding="application/x-tex">{\displaystyle d'(\mathbf {x} )=\mathbf {x} -{\frac {\alpha }{|\mathbf {x} |^{2}}}\mathbf {x} ,}</annotation>

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</math><img src="./1672c7502d63ce4a59524cd6654eaefb9e4a6327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.126ex; height:5.843ex;" alt="{\displaystyle d'(\mathbf {x} )=\mathbf {x} -{\frac {\alpha }{|\mathbf {x} |^{2}}}\mathbf {x} ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =n-2}">

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

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</math><img src="./16b69517a7c0a8b13b5ef9937e6f9a945d9052d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.984ex; height:2.343ex;" alt="{\displaystyle \alpha =n-2}" loading="lazy">. We will show that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'}">

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

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</math><img src="./1f310a68106a9e308bdaf887ff8f7171c4cb9d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}" loading="lazy"> is a better decision rule than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">. The risk function is

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<annotation encoding="application/x-tex">{\displaystyle R(\theta ,d')=\operatorname {E} _{\theta }\left[\left|\mathbf {\theta -X} +{\frac {\alpha }{|\mathbf {X} |^{2}}}\mathbf {X} \right|^{2}\right]}</annotation>

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</math><img src="./53af2890b7cd27b63bffa5f567c3e16c7debcedc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.017ex; height:8.176ex;" alt="{\displaystyle R(\theta ,d')=\operatorname {E} _{\theta }\left[\left|\mathbf {\theta -X} +{\frac {\alpha }{|\mathbf {X} |^{2}}}\mathbf {X} \right|^{2}\right]}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\operatorname {E} _{\theta }\left[|\mathbf {\theta -X} |^{2}+2(\mathbf {\theta -X} )^{T}{\frac {\alpha }{|\mathbf {X} |^{2}}}\mathbf {X} +{\frac {\alpha ^{2}}{|\mathbf {X} |^{4}}}|\mathbf {X} |^{2}\right]}</annotation>

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</math><img src="./d0d6bd94fd2a04861cc3855cc30155458f20913a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.373ex; height:7.509ex;" alt="{\displaystyle =\operatorname {E} _{\theta }\left[|\mathbf {\theta -X} |^{2}+2(\mathbf {\theta -X} )^{T}{\frac {\alpha }{|\mathbf {X} |^{2}}}\mathbf {X} +{\frac {\alpha ^{2}}{|\mathbf {X} |^{4}}}|\mathbf {X} |^{2}\right]}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle =\operatorname {E} _{\theta }\left[|\mathbf {\theta -X} |^{2}\right]+2\alpha \operatorname {E} _{\theta }\left[{\frac {\mathbf {(\theta -X)} ^{T}\mathbf {X} }{|\mathbf {X} |^{2}}}\right]+\alpha ^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]}</annotation>

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</math><img src="./2686f5f7cfc7cbf8533e1778676e5e7a8147404a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.789ex; height:7.509ex;" alt="{\displaystyle =\operatorname {E} _{\theta }\left[|\mathbf {\theta -X} |^{2}\right]+2\alpha \operatorname {E} _{\theta }\left[{\frac {\mathbf {(\theta -X)} ^{T}\mathbf {X} }{|\mathbf {X} |^{2}}}\right]+\alpha ^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]}" loading="lazy"></dd></dl>

— a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">. We may simplify the middle term by considering a general "well-behaved" function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:\mathbf {x} \mapsto h(\mathbf {x} )\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle h:\mathbf {x} \mapsto h(\mathbf {x} )\in \mathbb {R} }</annotation>

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</math><img src="./8050662a37c29f126cce33d42db3c2dbe4c8e430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.379ex; height:2.843ex;" alt="{\displaystyle h:\mathbf {x} \mapsto h(\mathbf {x} )\in \mathbb {R} }" loading="lazy"> and using <a href="Integration_by_parts" title="Integration by parts">integration by parts</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n}">

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</math><img src="./abbe58b9b83f8b6ec0da570e2249323a8930ef1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n}" loading="lazy">, for any continuously differentiable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> growing sufficiently slowly for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> we have:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }[(\theta _{i}-X_{i})h(\mathbf {X} )\mid X_{j}=x_{j}(j\neq i)]=\int (\theta _{i}-x_{i})h(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx_{i}}</annotation>

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</math><img src="./3be0db831ae8f4cc2317f7b827192c8752458d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:82.971ex; height:6.676ex;" alt="{\displaystyle \operatorname {E} _{\theta }[(\theta _{i}-X_{i})h(\mathbf {X} )\mid X_{j}=x_{j}(j\neq i)]=\int (\theta _{i}-x_{i})h(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx_{i}}" loading="lazy"></dd>

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</mrow>

<msup>

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</msup>

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</msup>

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<msub>

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<mfrac>

<mrow>

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<mrow>

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<msup>

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<mrow>

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<msup>

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</mrow>

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</msup>

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</msup>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\left[h(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}\right]_{x_{i}=-\infty }^{\infty }-\int {\frac {\partial h}{\partial x_{i}}}(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx_{i}}</annotation>

</semantics>

</math><img src="./045239e69b681a31bb764f1b76554b969bc68989.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:84.023ex; height:7.843ex;" alt="{\displaystyle =\left[h(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}\right]_{x_{i}=-\infty }^{\infty }-\int {\frac {\partial h}{\partial x_{i}}}(\mathbf {x} )\left({\frac {1}{2\pi }}\right)^{n/2}e^{-(1/2)({\boldsymbol {\theta }}-\mathbf {x} )^{T}({\boldsymbol {\theta }}-\mathbf {x} )}dx_{i}}" loading="lazy"></dd>

<msub>

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</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

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</msub>

</mrow>

</mfrac>

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</mrow>

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</msub>

<msub>

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</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =-\operatorname {E} _{\theta }\left[{\frac {\partial h}{\partial x_{i}}}(\mathbf {X} )\mid X_{j}=x_{j}(j\neq i)\right].}</annotation>

</semantics>

</math><img src="./d161b5f7363661735fc7dc2a16b7b2c8a7ad1ca0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.226ex; height:6.176ex;" alt="{\displaystyle =-\operatorname {E} _{\theta }\left[{\frac {\partial h}{\partial x_{i}}}(\mathbf {X} )\mid X_{j}=x_{j}(j\neq i)\right].}" loading="lazy"></dd></dl>

Therefore,

<msub>

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<msub>

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<msub>

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<mrow>

<mrow>

<mfrac>

<mrow>

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<mrow>

<msub>

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</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }[(\theta _{i}-X_{i})h(\mathbf {X} )]=-\operatorname {E} _{\theta }\left[{\frac {\partial h}{\partial x_{i}}}(\mathbf {X} )\right].}</annotation>

</semantics>

</math><img src="./cb61a83c56c891d9e294b55bea3335aeb8367e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.405ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} _{\theta }[(\theta _{i}-X_{i})h(\mathbf {X} )]=-\operatorname {E} _{\theta }\left[{\frac {\partial h}{\partial x_{i}}}(\mathbf {X} )\right].}" loading="lazy"></dd></dl>

(This result is known as <a href="Stein's_lemma" title="Stein's lemma">Stein's lemma</a>.) Now, we choose

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\mathbf {x} )={\frac {x_{i}}{|\mathbf {x} |^{2}}}.}</annotation>

</semantics>

</math><img src="./0f35dfd58111d51d8c4746b7e29ee64837197ed8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.9ex; height:5.843ex;" alt="{\displaystyle h(\mathbf {x} )={\frac {x_{i}}{|\mathbf {x} |^{2}}}.}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> met the "well-behaved" condition (it doesn't, but this can be remedied—see below), we would have

<mfrac>

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</mrow>

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial h}{\partial x_{i}}}={\frac {1}{|\mathbf {x} |^{2}}}-{\frac {2x_{i}^{2}}{|\mathbf {x} |^{4}}}}</annotation>

</semantics>

</math><img src="./a32703cb74e64f848f44e43a2a28e68dc027b94d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.412ex; height:7.176ex;" alt="{\displaystyle {\frac {\partial h}{\partial x_{i}}}={\frac {1}{|\mathbf {x} |^{2}}}-{\frac {2x_{i}^{2}}{|\mathbf {x} |^{4}}}}" loading="lazy"></dd></dl>

and so

<msub>

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<msup>

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</msup>

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</mfrac>

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</mrow>

</mrow>

</mrow>

</munderover>

<msub>

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</msub>

<mrow>

<mrow>

<msub>

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</msub>

<msub>

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</msub>

<mfrac>

<msub>

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</msub>

<mrow>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{\theta }\left[{\frac {({\boldsymbol {\theta }}-\mathbf {X} )^{T}\mathbf {X} }{|\mathbf {X} |^{2}}}\right]=\sum _{i=1}^{n}\operatorname {E} _{\theta }\left[(\theta _{i}-X_{i}){\frac {X_{i}}{|\mathbf {X} |^{2}}}\right]}</annotation>

</semantics>

</math><img src="./8247d0f67f4b234425004578b661708759bf452d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.122ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} _{\theta }\left[{\frac {({\boldsymbol {\theta }}-\mathbf {X} )^{T}\mathbf {X} }{|\mathbf {X} |^{2}}}\right]=\sum _{i=1}^{n}\operatorname {E} _{\theta }\left[(\theta _{i}-X_{i}){\frac {X_{i}}{|\mathbf {X} |^{2}}}\right]}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =-\sum _{i=1}^{n}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}-{\frac {2X_{i}^{2}}{|\mathbf {X} |^{4}}}\right]}</annotation>

</semantics>

</math><img src="./c03a081bb181a5391ce9f473d3bc091e3ca1fc24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.935ex; height:7.509ex;" alt="{\displaystyle =-\sum _{i=1}^{n}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}-{\frac {2X_{i}^{2}}{|\mathbf {X} |^{4}}}\right]}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =-(n-2)\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right].}</annotation>

</semantics>

</math><img src="./e4d9f21e647c8da339e4f087293ffd9ac32e6b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.002ex; height:7.509ex;" alt="{\displaystyle =-(n-2)\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right].}" loading="lazy"></dd></dl>

Then returning to the risk function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f310a68106a9e308bdaf887ff8f7171c4cb9d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}" loading="lazy">:

<msup>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(\theta ,d')=n-2\alpha (n-2)\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]+\alpha ^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right].}</annotation>

</semantics>

</math><img src="./3a406c5b50929abb1cffbf67a86c448e2340c5a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.595ex; height:7.509ex;" alt="{\displaystyle R(\theta ,d')=n-2\alpha (n-2)\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]+\alpha ^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right].}" loading="lazy"></dd></dl>

This quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is minimized at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =n-2}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./16b69517a7c0a8b13b5ef9937e6f9a945d9052d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.984ex; height:2.343ex;" alt="{\displaystyle \alpha =n-2}" loading="lazy">, giving

<msup>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(\theta ,d')=R(\theta ,d)-(n-2)^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]}</annotation>

</semantics>

</math><img src="./36aef6769e6264751d25a514511de5c4d6de1b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.601ex; height:7.509ex;" alt="{\displaystyle R(\theta ,d')=R(\theta ,d)-(n-2)^{2}\operatorname {E} _{\theta }\left[{\frac {1}{|\mathbf {X} |^{2}}}\right]}" loading="lazy"></dd></dl>

which of course satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(\theta ,d')<R(\theta ,d).}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(\theta ,d')<R(\theta ,d).}</annotation>

</semantics>

</math><img src="./3f908674b8ebaa1f03cc7db5deff53ee3ab7b436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.259ex; height:3.009ex;" alt="{\displaystyle R(\theta ,d')<R(\theta ,d).}" loading="lazy"> making <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> an inadmissible decision rule.

It remains to justify the use of

</mrow>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\mathbf {X} )={\frac {\mathbf {X} }{|\mathbf {X} |^{2}}}.}</annotation>

</semantics>

</math><img src="./b8378d7f3d0e3622919247244e41c6652aea0207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.117ex; height:6.509ex;" alt="{\displaystyle h(\mathbf {X} )={\frac {\mathbf {X} }{|\mathbf {X} |^{2}}}.}" loading="lazy"></dd></dl>

This function is not continuously differentiable, since it is singular at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =0}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e988dac229faa876a6324f3ef4b6e431e2db3b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle \mathbf {x} =0}" loading="lazy">. However, the function

</mrow>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(\mathbf {X} )={\frac {\mathbf {X} }{\varepsilon +|\mathbf {X} |^{2}}}}</annotation>

</semantics>

</math><img src="./9ebc5e40a3ae8fd9e08ea598a79ddcd9a6e6c486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.394ex; height:6.509ex;" alt="{\displaystyle h(\mathbf {X} )={\frac {\mathbf {X} }{\varepsilon +|\mathbf {X} |^{2}}}}" loading="lazy"></dd></dl>

is continuously differentiable, and after following the algebra through and letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \to 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f0a6823c23666f99317e232cf7d02df6d9c9b7a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.86ex; height:2.176ex;" alt="{\displaystyle \varepsilon \to 0}" loading="lazy">, one obtains the same result.

<ul><li><a href="James%E2%80%93Stein_estimator" title="James–Stein estimator">James–Stein estimator</a></li></ul>

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

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</style><cite id="CITEREFEfronMorris1977" class="citation cs2"><a href="Bradley_Efron" title="Bradley Efron">Efron, B.</a>; Morris, C. (1977), <a rel="nofollow" class="external text" href="http://www-stat.stanford.edu/~ckirby/brad/other/Article1977.pdf">"Stein's paradox in statistics"</a> (PDF), <a href="Scientific_American" title="Scientific American">Scientific American</a>, 236 (5): 119–127, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977SciAm.236e.119E">1977SciAm.236e.119E</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0577-119">10.1038/scientificamerican0577-119</a></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBrown1971" class="citation journal cs1"><a href="Lawrence_D._Brown" title="Lawrence D. Brown">Brown, L. D.</a> (1971). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.aoms/1177693318">"Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems"</a>. The Annals of Mathematical Statistics. 42 (3): 855–903. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177693318">10.1214/aoms/1177693318</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4851">0003-4851</a>.</cite>

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

<ul><li><cite id="CITEREFLehmannCasella1998" class="citation cs2"><a href="Erich_Leo_Lehmann" title="Erich Leo Lehmann">Lehmann, E. L.</a>; Casella, G. (1998), "ch.5", Theory of Point Estimation (2nd ed.), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-05849-1</bdi></cite></li>

<li><cite id="CITEREFStein1956" class="citation conference cs1"><a href="Charles_Stein_(statistician)" class="mw-redirect" title="Charles Stein (statistician)">Stein, C.</a> (1956). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bsmsp/1200501656">"Inadmissibility of the usual estimator for the mean of a multivariate distribution"</a>. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. pp. 197–206. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0084922">0084922</a>.</cite></li>

<li><cite id="CITEREFSamworth2012" class="citation cs2"><a href="Richard_Samworth" title="Richard Samworth">Samworth, R. J.</a> (2012), <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/~rjs57/SteinParadox.pdf">"Stein's Paradox"</a> (PDF), Eureka, 62: 38–41</cite></li></ul></div><div><div class="zim-footer">

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