Generalization of Sverdrup's Lemma and its Applications to Multivariate Distribution Theory

Abstract

The multivariate sampling distribution theory underlying a multivariate nornal law has a significant role in Multivariate Statistical Analysis. Several methods are available for the derivation of the usual sampling distributions, see, e.g., [2], [7], [8], [9], and [10]. This is yet another attempt in the same direction. However, the method which we present in this paper is elegant and straightforward, as the desired distributions are obtained in a rather unified way, either directly from the probability law of the sample or from the Wishart distribution. Our method is based on a generalization of Sverdrup's lemma [12] which we give below. The generalized Sverdrup's lemma is indeed implicit in many derivations in multivariate statistical literature without its explicit statement, see, e.g., Anderson's derivation of the integral representation of the noncentral Wishart distribution ([1], p. 417). Our purpose in this paper is to make an explicit statement of this implied lemma, and point out that the lemma stated here may be used as a powerful tool in multivariate distribution theory. Since our method is easily understood, we give only two applications. Several other applications follow on similar lines, see, Kabe [3], [4]. We have used fairly standard notation in this paper.