Asymptotic expansions for the joint distribution of correlated Hotelling's T2 statistics under normality

Let T-i(2) = z'S-i(-1) z(i), i = 1, ... , k be correlated Hotelling's T-2 statistics under normality, where z = (z'(1), ..., z'(k))' and nS are independently distributed as N-kp(0, Gamma x Sigma) and Wishart distribution W-p(Sigma, n), respectively. The purpose of this paper is to study the distribution function F(x(1), ..., x(k)) of (T-1(2), ... T-k(2)) when n is large. First we derive an asymptotic expansion of the characteristic function of (T-1(2), ..., T-k(2)) up to the order n(-2). Next we give asymptotic expansions for F(x(1), ... , x(k)) in two cases (i) Gamma = I-k and (ii) k = 2 by inverting the expanded characteristic function up to the orders n(-2) and n(-1), respectively. Our results can be applied to the distribution function of max(T-1(2), ... , T-k(2)) as a special case.