The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.

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Latent Roots and Matrix Variates: A Review of Some Asymptotic Results

Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices

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Asymptotic expansions for the distributions of statistics based on the sample correlation matrix in principal component analysis

An asymptotic expansion for the distributions of the latent roots of the Wishart matrix with multiple population roots

Partial differential equations for hypergeometric functions of complex argument matrices and their applications

: Some of the difficulties in multivariate analysis involve the calculation of moments and significance points of distributions, and of powers of tests. In many cases, due to the complexity of the distributions involved, exact determination of these quantities appears all but impossible, and it makes sense to look for approximate solutions. This thesis is concerned essentially with deriving asymptotic expansions associated with two particular latent root distributions.

Yasuko Chikuse

Papers

Asymptotic Expansions for the Distributions of the Latent Roots of Two Matrices in Multivariate Analysis.