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

Let be a pxp random matrix having a Wishart distribution . For testing a general covariance structure , we consider a class of test statistics , where is a distance measure from to , [lambda]i's are the eigenvalues of , 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 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. In particular, we investigate the power of the sphericity tests.