Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t² statlstics under normality

Abstract

Let T² _i=z′_iS⁻¹z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ⊗∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n⁻². Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=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 (T² _i,…,T² _k) as a special case.

Keywords:

• Asymptotic expansions
• Characteristic function
• carrelated Hoteling T²
• statistics Distributsion function max (T² _i,…,T² _k)