An Asymptotic Expansion of the Non-Null Distribution of Hotelling's Generalized $T_0^2$-Statistic

Abstract

Let $Z = \{z_1, z_2, \cdots, z_m\}$ be a $p \times m$ random matrix where $z_i$ are independently distributed according to $p$-variate normal distributions with means $\mu_i$ and common covariance matrix $\Lambda = (\lambda_{ij})(> 0$, positive definite) and let $nS_n = n(s_{ij})$ be a $p \times p$ matrix which is independent of $Z$ and is subject to a central Wishart distribution $W_p(\Lambda, n)$ with $n$ degrees of freedom and covariance matrix $\Lambda$. Hotelling's generalized $T_o^2$-statistic is then defined by \begin{equation*}\tag{1.1} T_o^2 = \operatorname{tr} S_n^{-1}ZZ' = \sum^m_{i=1} z_i'S_n^{-1}z_i\end{equation*} which is the statistic proposed by Hotelling [4], [5] for testing the hypothesis $H: M = \{\mu_1, \cdots, \mu_m\} = 0$ against $K: M \neq 0$. The distribution of $T_0^2$ when $H$ is true has been treated by several authors: for example, Hotelling [5], Ito [6], Siotani [13], [14], Pillai and Samson [12], and Davis [3]. Even in the null case, the exact distribution of $T_0^2$ is not available except for certain special values of $p$ and $m$. When $m = 1, T_0^2$ reduces to Hotelling's generalization of "Student's" $t$ and if $p = 1, (1/m)T_0^2$ is simply $F$-statistic. Hence the non-null distributions in these cases are known exactly. For the general non-null case, Constantine [2] has obtained the exact distribution as well as moments of $T_0^2/n$ using the generalized Laguerre polynomials of matrix argument. Unfortunately this distribution is valid only over the range $0 \leqq T_0^2/n < n 1$ and hence not so useful since we are usually interested in the upper tail of the distribution. Siotani [15] has treated an asymptotic expansion for the non-null distribution of $T_0^2$ according to the basic idea due to Welch [17] and James [9]. The same problem has been attacked by Ito [7], using the integral representation of the characteristic function of $T_0^2$. However formulas of these authors are inadequate for a good approximation to the distribution since they have only the terms up to order $n^{-1}$ (for some numerical information, see [8]) and also somewhat inconvenient terms for numerical work. Khatri and Pillai have evaluated the moments of their statistic $U^{(p)}$ (a constant times $T_0^2$) and given approximate distributions of $U^{(p)}$ (and hence of $T_0^2$) in the light of the first four general noncentral moments, the summary of which can be obtained in their recent paper [10]. In this paper an asymptotic expansion for the non-null distribution of $T_0^2$ is given up to the terms of order $n^{-2}$, in which the effect of the noncentrality is contained in the form $s_j = \operatorname{tr} \Omega^j, j = 1, 2, \cdots$ where $\Omega = \Lambda^{-1}MM'$. This is carried out by expanding the characteristic function of $T_0^2$ along the Welch-James idea and by inverting the resultant into the corresponding expansion of the distribution function or the density function.