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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 Generalized chi-squared distribution ： ウィキペディア英語版 Generalized chi-squared distribution In probability theory and statistics, the specific name generalized chi-squared distribution (also generalized chi-square distribution) arises in relation to one particular family of variants of the chi-squared distribution. There are several other such variants for which the same term is sometimes used, or which clearly are generalizations of the chi-squared distribution, and which are treated elsewhere: some are special cases of the family discussed here, for example the noncentral chi-squared distribution and the gamma distribution, while the generalized gamma distribution is outside this family. The type of generalisation of the chi-squared distribution that is discussed here is of importance because it arises in the context of the distribution of statistical estimates in cases where the usual statistical theory does not hold. For example, if a predictive model is fitted by least squares but the model errors have either autocorrelation or heteroscedasticity, then a statistical analysis of alternative model structures can be undertaken by relating changes in the sum of squares to an asymptotically valid generalized chi-squared distribution.〔Jones, D.A. (1983) "Statistical analysis of empirical models fitted by optimisation", Biometrika, 70 (1), 67–88〕 More specifically, the distribution can be defined in terms of a quadratic form derived from a multivariate normal distribution. ==Definition== One formulation of the generalized chi-squared distribution is as follows.〔 Let ''z'' have a multivariate normal distribution with zero mean and covariance matrix ''B'', then the value of the quadratic form ''X''=''z''TAz, where ''A'' is a matrix, has a generalised chi-squared distribution with parameters ''A'' and ''B''. Note that there is some redundancy in this formulation, as for any matrix ''C'', the distribution with parameters ''C''T''AC'' and ''B'' is identical to the distribution with parameters ''A'' and ''CBC''T. The most general form of generalized chi-squared distribution is obtained by extending the above consideration in two ways: firstly, to allow ''z'' to have a non-zero mean and, secondly, to include an additional linear combination of ''z'' in the definition of ''X''. Note that, in the above formulation, ''A'' and ''B'' need not be positive definite. However, the case where A is restricted to be at least positive semidefinite is an important one. For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:〔Sheil, J., O'Muircheartaigh, I. (1977) "Algorithm AS106: The distribution of non-negative quadratic forms in normal variables",''Applied Statistics'', 26, 92–98〕 :$X=(z+a)^\backslash mathrm\; T\; A(z+a)+c^\backslash mathrm\; T\; z=\; (x+b)^\backslash mathrm\; T\; D(x+b)+d^\backslash mathrm\; T\; x+e\; ,$ where ''D'' is a diagonal matrix and where ''x'' represents a vector of uncorrelated standard normal random variables. An alternative representation can be stated in the form:〔Davies, R.B. (1973) Numerical inversion of a characteristic function. Biometrika, 60 (2), 415–417〕〔Davies, R,B. (1980) "Algorithm AS155: The distribution of a linear combination of ''χ''2 random variables", ''Applied Statistics'', 29, 323–333〕 :$X=\backslash sum\_^r\; \backslash lambda\_i\; Y\_i\; +\; f\; Z\_0\; ,$ where the ''Yi'' represent random variables having (different) noncentral chi-squared distributions, where ''Z''0 has a standard normal distribution, and where all these random variables are independent. Some important special cases relating to this particular form either omit the additional standard normal term and/or have central rather than non-central chi-squared distributions for the components of the summation. 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「Generalized chi-squared distribution」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.