On Multivariate Distribution Theory

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Abstract
This paper is concerned with a matrix method of deriving the sampling distributions of a large class of statistics directly from the probability law for random samples from a multivariate normal population, that is without assuming the Wishart distribution or the distribution of rectangular coordinates. Two techniques are proposed for evaluating the Jacobians of certain transformations, one based on a theorem on Jacobians [1], and the second based on the introduction of pseudo or extra variables. This matrix approach has a geometrical analog developed in part by one of the authors [2]. Section 3 is concerned with a discussion of these two techniques; in Section 4, the former is applied to obtain the joint distribution of the rectangular coordinates [3], and in Section 5, the second method is applied to obtain the joint distribution of the roots of a determinantal equation [4], [5], [6], and [7].

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