Asymptotic expansions for a class of tests for a general covariance structure under a local alternative

JOURNAL OF MULTIVARIATE ANALYSIS

卷:102

Article

Shimizu, Hiroaki¹  Wakaki, Hirofumi¹ 

[1] Hiroshima Univ, Dept Math, Higashihiroshima 7398526, Japan

关键词: Asymptotic expansion;    Class of test statistics;    General covariance structure;    Non-null distribution;    Local alternative;    Power comparison;    Linear structure;    Sphericity test;   

DOI  :  10.1016/j.jmva.2011.03.002

来源: Elsevier

【 摘 要 】

Let S be a p x p random matrix having a Wishart distribution W-p(n, n(-1) Sigma). For testing a general covariance structure Sigma = Sigma(xi), we consider a class of test statistics T-h = n rho(h)(S, Sigma((xi) over cap)), where rho(h) (Sigma(1), Sigma(2)) = Sigma(p)(i=1) h(lambda(i)) is a distance measure from Sigma(1) to Sigma(2), lambda(i)'s are the eigenvalues of Sigma(1)Sigma(-1)(2), and his a given function with certain properties. Wakaki, Eguchi and Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of T-h. This paper gives an asymptotic expansion of the non-null distribution of T-h under a sequence of alternatives. By using results, we derive the power, and compare the power asymptotically in the class. In particular, we investigate the power of the sphericity tests. (C) 2011 Elsevier Inc. All rights reserved.

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