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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 Eaton's inequality ： ウィキペディア英語版 Eaton's inequality In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. This inequality was described in 1974 by Morris L. Eaton.〔Eaton, Morris L. (1974) "A probability inequality for linear combinations of bounded random variables." ''Annals of Statistics'' 2(3) 609–614〕 ==Statement of the inequality== Let ''X''i be a set of real independent random variables, each with a expected value of zero and bounded by 1 ( | ''X''''i'' | ≤ 1, for 1 ≤ ''i'' ≤ ''n''). The variates do not have to be identically or symmetrically distributed. Let ''a''''i'' be a set of ''n'' fixed real numbers with : $\backslash sum\_^n\; a\_i^2\; =\; 1\; .$ Eaton showed that : $P\backslash left(\; \backslash left|\; \backslash sum\_^n\; a\_i\; X\_i\; \backslash right|\; \backslash ge\; k\; \backslash right)\; \backslash le\; 2\; \backslash inf\_\; \backslash int\_c^\backslash infty\; \backslash left(\; \backslash frac\; \backslash right)^3\; \backslash phi(\; z\; )\; \backslash ,\; dz\; =\; 2\; B\_E(\; k\; )\; ,$ where ''φ''(''x'') is the probability density function of the standard normal distribution. A related bound is Edelman's : $P\backslash left(\; \backslash left|\; \backslash sum\_^n\; a\_i\; X\_i\; \backslash right|\; \backslash ge\; k\; \backslash right)\; \backslash le\; 2\; \backslash left(\; 1\; -\; \backslash Phi\backslash left(k\; -\; \backslash frac\; \backslash right)\; \backslash right)\; =\; 2\; B\_(\; k\; )\; ,$ where Φ(''x'') is cumulative distribution function of the standard normal distribution. Pinelis has shown that Eaton's bound can be sharpened:〔Pinelis, I. (1994) "Extremal probabilistic problems and Hotelling's ''T''2 test under a symmetry condition." ''Annals of Statistics'' 22(1), 357–368〕 : $B\_\; =\; \backslash min\backslash$ A set of critical values for Eaton's bound have been determined.〔Dufour, J-M; Hallin, M (1993) "Improved Eaton bounds for linear combinations of bounded random variables, with statistical applications", ''Journal of the American Statistical Association'', 88(243) 1026–1033〕 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「Eaton's inequality」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.