A System of Linear Differential Equations for the Distribution of Hotelling's Generalized $T_o^2$

Abstract

Let $\mathscr{S}_1, \mathscr{S}_2$ be independent $m \times m$ matrices on $n_1, n_2$ degrees of freedom respectively, $\mathscr{S}_2$ having a Wishart distribution and $\mathscr{S}_1$ having a possibly non-central Wishart distribution with the same covariance matrix. Hotelling's generalized $T_0^2$ statistic is then defined [7] by \begin{equation*}\tag{1.1}T = n^{-1}_2T_0^2 = \operatorname{tr} \mathscr{S}_1\mathscr{S}^{-1}_2.\end{equation*} The complete distribution of this statistic is known only in particular cases. If $m = 1,$ then $(n_2/n_1)T$ is simply non-central $F$. In the case $n_1 = 1, T$ reduces to Hotelling's generalization of "Student's" $t,$ which also has a non-central $F$ distribution. When $m = 2$, Hotelling [7] has shown that in the null case the density function of $T$ is \begin{equation*}\tag{1.2}f(T) = \lbrack\Gamma(n_1 + n_2 - 1)/\Gamma(n_1)\Gamma (n_2 - 1)\rbrack(\frac{1}{2}T)^{n_1-1}(1 + \frac{1}{2}T)^{-(n_1+n_2)} \cdot_2F_1 (1, \frac{1}{2}(n_1 + n_2); \frac{1}{2}(n_2 + 1); v),\end{equation*} where $\nu = T^2/(T + 2)^2$, and $_2F_1$ is the Gaussian hypergeometric function. When $n_2$ becomes large, the distribution of $T_0^2$ approaches that of $\chi^2$ based on $mn_1$ degrees of freedom. Ito [9] has derived asymptotic expansions both for the cumulative distribution function (cdf) of $T_0^2,$ and for the percentiles of $T_0^2$ in terms of the corresponding $\chi^2_{mn_1}$ percentiles. Other approximations to the distribution requiring large $n_2$ for validity have been obtained by Pillai and Samson [12]. These authors have used the method of fitting a Pearson curve by means of moment quotients to tabulate upper 5% and 1% points for $m = 2, 3, 4$. The exact distribution of $T$ over the range $0 \leqq T < 1$ has been obtained in the general non-central case by Constantine [3], using the methods of zonal polynomials and hypergeometric functions of matrix argument developed by James and Constantine ([2] and [10], for example). Constantine's solution has the form \begin{equation*}\tag{1.3}f(T) = \lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack T ^{\frac{1}{2}mn_1-1}\mathscr{P}(T),\end{equation*} where $\mathscr{P}(T)$ is a power series in $T$ convergent in the unit circle, and \begin{equation*}\tag{1.4}\Gamma_m(z) = \pi^{\frac{1}{4}m(m-1)} \prod^{m-1}_{i=0} \Gamma(z - \frac{1}{2}i).\end{equation*} In Section 2 of the present paper, it is shown that in the null case the density function $f(T)$ (or rather, its analytic continuation into the complex $T$-plane) satisfies an ordinary linear differential equation of degree $m$ of Fuchsian type, having regular singularities at $T = 0, -1, \cdots, -m$ and infinity. More specifically, an equivalent first-order system is obtained, and the problem is most conveniently treated in this form. Constantine's series (1.3) in the null case is shown in Section 3 to be the relevant solution for $f(T)$ in the neighborhood of the regular singularity at $T = 0$. The differential equations lead to convenient recurrence relations for the coefficients in $\mathscr{P}(T)$. In Section 4 an alternative derivation of Ito's asymptotic formula is presented. Preliminary results are then given (Section 5) for the regular singularity at $T = \infty$, and a heuristic treatment of the limiting distribution as $n_1 \rightarrow \infty$ is presented in Section 6. Finally, it is shown in Section 7 that the moments of $T$ may be obtained from the differential equations for the Laplace transform of $f(T)$ given in Section 1. One objective in deriving the differential equations for $f(T)$ has been to obtain a convenient exact method for computing the distribution and its percentiles. This work is in progress, and it is hoped that results will be available shortly.