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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 Multivariate Behrens–Fisher problem ： ウィキペディア英語版 Multivariate Behrens–Fisher problem In statistics, the multivariate Behrens–Fisher problem is the problem of testing for the equality of means from two multivariate normal distributions when the covariance matrices are unknown and possibly not equal. Since this is a generalization of the univariate Behrens-Fisher problem, it inherits all of the difficulties that arise in the univariate problem. ==Notation and problem formulation== Let $X\_\; \backslash sim\; \backslash mathcal\_p(\backslash mu\_i,\backslash ,\; \backslash Sigma\_i)\; \backslash \; \backslash \; (j=1,\backslash dots,n\_i;\; \backslash \; \backslash \; i=1,2)\backslash$ be independent random samples from two $p$-variate normal distributions with unknown mean vectors $\backslash mu\_i$ and unknown dispersion matrices $\backslash Sigma\_i$. The index $i$ refers to the first or second population, and the $j$th observation from the $i$th population is $X\_$. The multivariate Behrens–Fisher problem is to test the null hypothesis $H\_0$ that the means are equal versus the alternative $H\_1$ of non-equality: : $H\_0\; :\; \backslash mu\_1\; =\; \backslash mu\_2\; \backslash \; \backslash \; \backslash text\; \backslash \; \backslash \; H\_1\; :\; \backslash mu\_1\; \backslash neq\; \backslash mu\_2.$ Define some statistics, which are used in the various attempts to solve the multivariate Behrens–Fisher problem, by : $$ \begin \bar &= \frac \sum_^ X_, \\ A_i &= \sum_^ (X_ - \bar)(X_ - \bar)', \\ S_i &= \frac A_i, \\ \tilde &= \fracS_i, \\ \tilde &= \tilde + \tilde, \quad \text \\ T^2 & = (\bar - \bar)'\tilde^(\bar - \bar). \end The sample means $\backslash bar$ and sum-of-squares matrices $A\_i$ are sufficient for the multivariate normal parameters $\backslash mu\_i,\; \backslash Sigma\_i,\backslash \; (i=1,2)$, so it suffices to perform inference be based on just these statistics. The distributions of $\backslash bar$ and $A\_i$ are independent and are, respectively, multivariate normal and Wishart:〔 : $$ \begin \bar &\sim \mathcal_p \left(\mu_i, \Sigma_i/n_i \right), \\ A_i &\sim W_p(\Sigma_i, n_i - 1). \end 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「Multivariate Behrens–Fisher problem」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.