On the Distribution of the Sample Covariance from a Matrix Normal Population

TL;DR

The paper addresses the joint distribution of sample variances and covariances, expressed as quadratic forms in a matrix population, in a multi-group setting with hypotheses $H_0: M^{(1)} = \cdots = M$. It classifies the $N = np$ intra-group variances and $\tfrac{1}{2}N(N-1)$ intra-group covariances according to four spectral forms of the precision matrix ($T_1$, $T_{1/2}$, $T_2$, $T_3$) and derives their joint distribution, along with the moment generating function and the joint distribution of latent roots. The work discusses applications to noncentral means with known covariance (e.g., one-sample ANOVA) and discriminant analysis where intra-group covariances differ, and also analyzes the distribution of the ratio of two quadratic forms in both central and non-central cases with exact power tabulations for varying $n$ and $p$. These results provide exact distributions and powers for related multivariate tests in matrix-normal settings.

Abstract

This paper discusses the joint distribution of sample variances and covariances, expressed in quadratic forms in a matrix population arising in comparing the differences among groups under homogeneity of variance. One major concern of this article is to compare $K$ different populations, by assuming that the mean values of $x_{11}^{(k)}, x_{12}^{(k)}, \dots, x_{1p}^{(k)}, x_{21}^{(k)}, x_{22}^{(k)}, \dots$, $x_{2p}^{(k)},\dots, x_{n1}^{(k)},x_{n2}^{(k)},\dots,$ $x_{np}^{(k)}$ in each population are $M^{(k)}$ ($n\times p$), $k = 1,2,\dots,K$ and $M$($n\times p$) a fixed matrix, with this hypothesis $$H_0: M^{(1)} = M^{(2)} = \dots = M^{(k)} = M,$$ when the inter-group covariances are neglected and the intra-group covariances are equal. The $N$ intra-group variances and $\frac{1}{2} N (N - 1)$ intra-group covariances where $N = np$ are classified into four categories $T_{1}$, $T_{1\frac{1}{2}}$, $T_{2}$ and $T_{3}$ according to the spectral forms of the precision matrix. The joint distribution of the sample variances and covariances is derived under these four scenarios. Besides, the moment generating function and the joint distribution of latent roots are explicitly calculated. %The distribution of non-central means with known covariance is calculated as an application to the one-sample analysis of variance, with its exact power tabulated up to order two. As an application, we consider a classification problem in the discriminant analysis where the two populations should have different intra-group covariances. The distribution of the ratio of two quadratic forms is considered both in the central and non-central cases, with their exact power tabulated for different $n$ and $p$.