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

Let S be a p×pp×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(ξˆ)), where ρh(Σ1,Σ2)=∑i=1ph(λi) is a distance measure from Σ1 to Σ2, λiλi’s are the eigenvalues of Σ1Σ2−1, and hh is a given function with certain properties. Wakaki, Eguchi and Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of ThTh. This paper gives an asymptotic expansion of the non-null distribution of ThTh 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.