The Annals of Mathematical Statistics

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

A. W. Davis

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  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  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  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 . 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 , using the methods of zonal polynomials and hypergeometric functions of matrix argument developed by James and Constantine ( and , 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.

Article information

Source
Ann. Math. Statist., Volume 39, Number 3 (1968), 815-832.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698313

Digital Object Identifier
doi:10.1214/aoms/1177698313

Mathematical Reviews number (MathSciNet)
MR225407

Zentralblatt MATH identifier
0257.62033

JSTOR
Davis, A. W. A System of Linear Differential Equations for the Distribution of Hotelling's Generalized $T_o^2$. Ann. Math. Statist. 39 (1968), no. 3, 815--832. doi:10.1214/aoms/1177698313. https://projecteuclid.org/euclid.aoms/1177698313