Asymptotic formulae for the distributions of some criteria for tests of equality of covariance matrices

The asymptotic expansions are derived up to terms of order 1/n, for the c.d.f. and percentile of the statistic T = m Tr S1S2-1, where mS1 and nS2 are independently distributed W(m, p, [Sigma]1) and W(n, p, [Sigma]2), respectively. The expansions hold when [Sigma]1[Sigma]2-1 = I + E and Chi(E) < 1. This technique is utilized to find the asymptotic expansion for the c.d.f. and percentile of F' = (m1/n1)(Tr S1S4-1/Tr S3S2-1), where m1S1, m2S2, n1S3, n2S4 are independently distributed Wishart matrices with degrees of freedom m1, m2, n1, n2, respectively, and each of the pairs (S1, S2), (S3, S4) has a common Covariance matrix. Finally, an asymptotic expansion for the F'max in a special case has been attempted where F'max is a criterion suggested for tests of equality of common Covariance matrices of the pairs above [6].