Asymptotic joint distribution of the largest roots of several multivariate F matrices I. Quasi-independent case

An asymptotic expansion of the joint distribution of k largest characteristic roots C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is given, where S'_is and S₀ are independent Wishart matrices with common covariance matrix Σ. The modified second-approximation procedure to the upper percentage points of the maximum of C_M⁽ⁱ⁾(S_iS₀^?1), i = 1,…, k, is also considered. The evaluation of the expansion is based on the idea for studentization due to Welch and James with the use of differential operators and of the perturbation procedure.