The Distribution of Hotelling's Generalised $T_0^2$

Abstract

The $T_0^2$-statistic was introduced by Hotelling [5], [6] as a measure of multivariate dispersion in connection with the problem of testing the accuracy of bombsights. A similar statistic was considered earlier by Lawley [10] as a generalisation of the $F$-test for testing significance in multivariate analyses of variance. In general, if the $m \times m$ matrices $S_1$ and $S_2$ are independently distributed on $n_1$ and $n_2$ degrees of freedom respectively, estimating the same covariance matrix, then $T_0^2$ is defined by $T = T_0^2/n_2 = \mathrm{tr} S_1S^{-1}_2.$ For instance, in a one-way classification analysis of variance $S_1$ and $S_2$ may be the "between classes" and "within classes" matrices of sums of squares and products. The matrix $S_1$ may be singular, i.e., $n_1 < m$, but $S_2$ is assumed non-singular. Assuming that one is sampling from a normal population, $S_2$ has the Wishart distribution and $S_1$ the (possibly) non-central Wishart distribution if $n_1 \geqq m$. If $n_1 < m$, the distribution of $T$ may be obtained from that for $n_1 \geqq m$ by a simple substitution (see Section 4). Hotelling derived the distribution of $T$ when $n_1 = 1$ (in which case $T$ is the generalisation of "Student's" $t$ [4]) and for $m = 2$, [6]. In Section 4, the distribution of $T$ will be derived for arbitrary $m, n_1$ and $n_2$ in the non-central case. More precisely, if $\Omega$ is the matrix of non-centrality parameters, the probability density function of $T$ has the series expansion \begin{equation*}\tag{1}\lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack e^{\mathrm{tr}(-\Omega)}T^{\frac{1}{2}mn_1-1}\end{equation*} $\cdot \sum^\infty_{k = 0}\lbrack(-T)^k/(\frac{1}{2}mn_1)_kk!\rbrack\sum_\kappa (\frac{1}{2}(n_1 + n_2))_\kappa L^\gamma_\kappa(\Omega),$ $\gamma = \frac{1}{2}(n_1 - m - 1),\quad |T| < 1.$ The functions $L^\gamma_\kappa (\Omega)$ are polynomials in the elements of $\Omega$ and are extensions of the classical Laguerre polynomials, to which they reduce when $m = 1$. They will be defined and studied in Section 3. The constants and coefficients occurring in the series are defined in Section 2. If $\Omega = 0$, the density function is \begin{equation*}\tag{2}\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}\end{equation*} $\cdot \sum^\infty_{k = 0}\lbrack (-T)^k/(\frac{1}{2}mn_1)_kk!\rbrack \sum_\kappa (\frac{1}{2}n_1)_\kappa (\frac{1}{2}(n_1 + n_2))_\kappa C_\kappa(I),$ where $C_\kappa(I)$ is the zonal polynomial evaluated at the identity matrix (see Section 2). Both series (1) and (2) converge only for $\|T\| < 1$. In the case $m = 1$, the series in (2) reduces to the binomial series for $(1 + T)^{-\frac{1}{2} (n_1+n_2)}$. Unfortunately, this is not very useful since one is usually interested in the upper tail of the distribution. However, it is hoped that the series may be simplified or be the basis for further studies of the distributions. In Section 5, the moments of $T$ are given, again in terms of the generalised Laguerre polynomials in the non-central case. As in the case of the $F$-distribution, only moments of sufficiently low order exist.