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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Proof of Stein's example ： ウィキペディア英語版 Proof of Stein's example Stein's example is an important result in decision theory which can be stated as : ''The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk in dimension at least 3''. The following is an outline of its proof. The reader is referred to the main article for more information. ==Sketched proof== The risk function of the decision rule $d(\backslash mathbf)\; =\; \backslash mathbf$ is :$R(\backslash theta,d)\; =\; \backslash mathbb\_\backslash theta(|\backslash mathbf|^2)$ ::$=\backslash int\; (\backslash mathbf)^T(\backslash mathbf)\; \backslash left(\; \backslash frac\; \backslash right)^\; e^)\; \}\; m(dx)$ ::$=\; n.\backslash ,$ Now consider the decision rule :$d\text{'}(\backslash mathbf)\; =\; \backslash mathbf\; -\; \backslash frac\backslash mathbf$ where $\backslash alpha\; =\; n-2$. We will show that $d\text{'}$ is a better decision rule than $d$. The risk function is :$R(\backslash theta,d\text{'})\; =\; \backslash mathbb\_\backslash theta\backslash left(\backslash left|\backslash mathbf\; +\; \backslash frac\backslash mathbf\backslash right|^2\backslash right)$ ::$=\; \backslash mathbb\_\backslash theta\backslash left(|\backslash mathbf|^2\; +\; 2(\backslash mathbf)^T\backslash frac\backslash mathbf\; +\; \backslash frac|\backslash mathbf|^2\; \backslash right)$ :: — a quadratic in $\backslash alpha$. We may simplify the middle term by considering a general "well-behaved" function $h:\backslash mathbf\; \backslash mapsto\; h(\backslash mathbf)\; \backslash in\; \backslash mathbb$ and using integration by parts. For $1\backslash leq\; i\; \backslash leq\; n$, for any continuously differentiable $h$ growing sufficiently slowly for large $x\_i$ we have: ::$\backslash mathbb\_\backslash theta\; ((\backslash theta\_i\; -\; X\_i)\; h(\backslash mathbf)\; |\; X\_j=x\_j\; (j\backslash neq\; i))=\; \backslash int\; (\backslash theta\_i\; -\; x\_i)\; h(\backslash mathbf)\; \backslash left(\; \backslash frac\; \backslash right)^\; e^\; \}\; m(dx\_i)$ :$=\; \backslash left(h(\backslash mathbf)\; \backslash left(\; \backslash frac\; \backslash right)^\; e^\; \}\; \backslash right)^\backslash infty\_$ - \int \frac(\mathbf) \left( \frac \right)^ e^ } m(dx_i) :$=\; -\; \backslash mathbb\_\backslash theta\; \backslash left(\backslash frac(\backslash mathbf)\; |\; X\_j=x\_j\; (j\backslash neq\; i)\; \backslash right).$ Therefore, :$\backslash mathbb\_\backslash theta\; ((\backslash theta\_i\; -\; X\_i)\; h(\backslash mathbf))=\; -\; \backslash mathbb\_\backslash theta\; \backslash left(\backslash frac(\backslash mathbf)\; \backslash right).$ (This result is known as Stein's lemma.) Now, we choose :$$ h(\mathbf) = \frac. If $h$ met the "well-behaved" condition (it doesn't, but this can be remedied -- see below), we would have :$\backslash frac\; =\; \backslash frac\; -\; \backslash frac$ and so ::$$ \mathbb_\theta\left((\theta_i - X_i) \frac \right ) :$=\; -\; \backslash sum\_^n\; \backslash mathbb\_\backslash theta\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)$ :$=\; -(n-2)\backslash mathbb\_\backslash theta\; \backslash left().$ Then returning to the risk function of $d\text{'}$ : :$$ R(\theta,d') = n - 2\alpha(n-2)\mathbb_\theta\left() + \alpha^2\mathbb_\theta\left(\right ). This quadratic in $\backslash alpha$ is minimized at :$\backslash alpha\; =\; n-2,\backslash ,$ giving :$R(\backslash theta,d\text{'})\; =\; R(\backslash theta,d)\; -\; (n-2)^2\backslash mathbb\_\backslash theta\backslash left(\backslash right)$ which of course satisfies: :$$ R(\theta,d') < R(\theta,d). making $d$ an inadmissible decision rule. It remains to justify the use of :$$ h(\mathbf)= \frac|^2}. This function is not continuously differentiable since it is singular at $\backslash mathbf=0$. However the function :$$ h(\mathbf) = \frac|^2} is continuously differentiable, and after following the algebra through and letting $\backslash epsilon\; \backslash to\; 0$ one obtains the same result. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proof of Stein's example」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.