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

Let S be a p × p random matrix having a Wishart distribution Wp(n, n−1Σ). For testing a general covariance structure Σ = Σ(ξ), we consider a class of test statistics Th = nρh(S, Σ(ˆξ)), where ρh(Σ1, Σ2) = p i=1 h(λi) is a distance measure from Σ1 to Σ2, λi’s are the eigenvalues of Σ1Σ−1 2 , and h is a given function with certain properties. Wakaki, Eguchi, Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of Th. This paper gives an asymptotic expansion of the non-null distribution of Th under a sequence of alternatives. By using results, we derive the power, and compare the power asymptotically in the class. Especially we investigate the power of the sphericity tests. 1 

• Type: Journal
• Year: 2011
• Where: MA
• Authors: Hiroaki Shimizu, Hirofumi Wakaki
• Keywords: Asymptotic Expansion, Communications, General Covariance Structure, MA 2011, Test Statistics Th