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 W p ( n , n − 1 Σ ) . For testing a general covariance structure Σ = Σ ( ξ ) , we consider a class of test statistics T h = n ρ h ( S , Σ ( ξ ˆ ) ) , where ρ h ( Σ 1 , Σ 2 ) = ∑ i = 1 p h ( λ i ) is a distance measure from Σ 1 to Σ 2 , λ i ’s are the eigenvalues of Σ 1 Σ 2 − 1 , and h 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 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.