On Multivariate Distribution Theory

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.