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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 Neyman–Pearson lemma ： ウィキペディア英語版 Neyman–Pearson lemma In statistics, the Neyman–Pearson lemma, named after Jerzy Neyman and Egon Pearson, states that when performing a hypothesis test between two simple hypotheses ''H''0: ''θ'' = ''θ''0 and ''H''1: ''θ'' = ''θ''1, the likelihood-ratio test which rejects ''H''0 in favour of ''H''1 when :$\backslash Lambda(x)=\backslash frac\; \backslash leq\; \backslash eta$ where :$P(\backslash Lambda(X)\backslash leq\; \backslash eta\backslash mid\; H\_0)=\backslash alpha$ is the most powerful test at significance level ''α'' for a threshold η. If the test is most powerful for all $\backslash theta\_1\; \backslash in\; \backslash Theta\_1$, it is said to be uniformly most powerful (UMP) for alternatives in the set $\backslash Theta\_1\; \backslash ,$. In practice, the likelihood ratio is often used directly to construct tests — see Likelihood-ratio test. However it can also be used to suggest particular test-statistics that might be of interest or to suggest simplified tests — for this, one considers algebraic manipulation of the ratio to see if there are key statistics in it related to the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one). == Proof == Define the rejection region of the null hypothesis for the NP test as :$R\_=\backslash left\backslash \; \backslash leq\; \backslash eta\backslash right\backslash \}$ where $\backslash eta$ is chosen so that $P(R\_,\backslash theta\_0)=\backslash alpha\backslash ,$. Any other test will have a different rejection region that we define as $R\_A$. Furthermore, define the probability of the data falling in region R, given parameter $\backslash theta$ as :$P(R,\backslash theta)=\backslash int\_R\; L(\backslash theta|x)\backslash ,\; dx,$ For the test with critical region $R\_A$ to have level $\backslash alpha$, it must be true that $\backslash alpha\; \backslash ge\; P(R\_A,\; \backslash theta\_0)$, hence :$\backslash alpha=\; P(R\_,\; \backslash theta\_0)\; \backslash ge\; P(R\_A,\; \backslash theta\_0)\; \backslash ,.$ It will be useful to break these down into integrals over distinct regions: :$P(R\_,\backslash theta)\; =\; P(R\_\; \backslash cap\; R\_A,\; \backslash theta)\; +\; P(R\_\; \backslash cap\; R\_A^c,\; \backslash theta),$ and :$P(R\_A,\backslash theta)\; =\; P(R\_\; \backslash cap\; R\_A,\; \backslash theta)\; +\; P(R\_^c\; \backslash cap\; R\_A,\; \backslash theta).$ Setting $\backslash theta=\backslash theta\_0$, these two expressions and the above inequality yield that :$P(R\_\; \backslash cap\; R\_A^c,\; \backslash theta\_0)\; \backslash ge\; P(R\_^c\; \backslash cap\; R\_A,\; \backslash theta\_0).$ Comparing the powers of the two tests, $P(R\_,\backslash theta\_1)$ and $P(R\_A,\backslash theta\_1)$, one can see that :$P(R\_,\backslash theta\_1)\; \backslash geq\; P(R\_A,\backslash theta\_1)\; \backslash iff$ P(R_ \cap R_A^c, \theta_1) \geq P(R_^c \cap R_A, \theta_1). Now by the definition of $R\_$, :$P(R\_\; \backslash cap\; R\_A^c,\; \backslash theta\_1)=\; \backslash int\_\; L(\backslash theta\_|x)\backslash ,dx\; \backslash geq\; \backslash frac\; \backslash int\_\; L(\backslash theta\_0|x)\backslash ,dx\; =\; \backslash fracP(R\_\; \backslash cap\; R\_A^c,\; \backslash theta\_0)$ :$\backslash ge\; \backslash fracP(R\_^c\; \backslash cap\; R\_A,\; \backslash theta\_0)\; =\; \backslash frac\backslash int\_\; L(\backslash theta\_|x)\backslash ,dx\; \backslash geq\; \backslash int\_\; L(\backslash theta\_|x)dx\; =\; P(R\_^c\; \backslash cap\; R\_A,\; \backslash theta\_1).$ Hence the inequality holds. 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「Neyman–Pearson lemma」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.