A class of tests for a general covariance structure

Let S be a p - p random matrix having a Wishart distribution Wp(n,n-1[Sigma]). For testing a general covariance structure [Sigma] = [Sigma]([xi]), we consider a class of test statistics Th = n inf [varrho]h(S, [Sigma]([xi])), where [varrho]h([Sigma]1, [Sigma]2) = [Sigma]j = 1ph([lambda]j) is a distance measure from [Sigma]1 to [Sigma]2, [lambda]i's are the eigenvalues of [Sigma]1[Sigma]2-1, and h is a given function with certain properties. This paper gives an asymptotic expansion of the null distribution of Th up to the order n-1. Using the general asymptotic formula, we give a condition for Th to have a Bartlett adjustment factor. Two special cases are considered in detail when [Sigma] is a linear combination or [Sigma]-1 is a linear combination of given matrices.