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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 UMVU ： ウィキペディア英語版 Minimum-variance unbiased estimator In statistics a uniformly minimum-variance unbiased estimator or minimum-variance unbiased estimator (UMVUE or MVUE) is an unbiased estimator that has ''lower variance'' than ''any other'' unbiased estimator for ''all possible'' values of the parameter. For practical statistics problems, it is important to determine the UMVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While the particular specification of "optimal" here — requiring unbiasedness and measuring "goodness" using the variance — may not always be what is wanted for any given practical situation, it is one where useful and generally applicable results can be found. ==Definition== Consider estimation of $g(\backslash theta)$ based on data $X\_1,\; X\_2,\; \backslash ldots,\; X\_n$ i.i.d. from some member of a family of densities $p\_\backslash theta,\; \backslash theta\; \backslash in\; \backslash Omega$, where $\backslash Omega$ is the parameter space. An unbiased estimator $\backslash delta(X\_1,\; X\_2,\; \backslash ldots,\; X\_n)$ of $g(\backslash theta)$ is ''UMVUE'' if $\backslash forall\; \backslash theta\; \backslash in\; \backslash Omega$, :$\backslash mathrm(\backslash delta(X\_1,\; X\_2,\; \backslash ldots,\; X\_n))\; \backslash leq\; \backslash mathrm(\backslash tilde(X\_1,\; X\_2,\; \backslash ldots,\; X\_n))$ for any other unbiased estimator $\backslash tilde.$ If an unbiased estimator of $g(\backslash theta)$ exists, then one can prove there is an essentially unique MVUE. Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding a complete sufficient statistic for the family $p\_\backslash theta,\; \backslash theta\; \backslash in\; \backslash Omega$ and conditioning ''any'' unbiased estimator on it. Further, by the Lehmann–Scheffé theorem, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVUE estimator. Put formally, suppose $\backslash delta(X\_1,\; X\_2,\; \backslash ldots,\; X\_n)$ is unbiased for $g(\backslash theta)$, and that $T$ is a complete sufficient statistic for the family of densities. Then :$\backslash eta(X\_1,\; X\_2,\; \backslash ldots,\; X\_n)\; =\; \backslash mathrm(\backslash delta(X\_1,\; X\_2,\; \backslash ldots,\; X\_n)|T)\backslash ,$ is the MVUE for $g(\backslash theta).$ A Bayesian analog is a Bayes estimator, particularly with minimum mean square error (MMSE). 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「Minimum-variance unbiased estimator」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.