On distributions of covariance structures

A. M. Mathai and Nicy Sebastian

Communications in Statistics - Theory and Methods, 2023, vol. 52, issue 20, 7370-7384

Abstract: The derivation of the sample covariance is difficult compared to that of the distribution of the sample correlation coefficient. This paper deals with the distributions of covariance structures appearing in real scalar/vector/matrix variables. Covariance structure is a bilinear structure. Consider a bilinear form u=X′AY where X and Y are p×1 and q×1 real vectors and A is a constant p × q matrix. The basic aim in this paper is to derive the distribution of such a structure when the components are scalar/vector/matrix Gaussian variables. The procedure used is to examine the Laplace transform or the moment generating function (mgf) coming from such a bilinear form in real scalar/vector/matrix variables. Covariance structures in several situations are shown to produce a mgf of the type (1−λ2t2)−α,λ>0,α>0,−1λ

Date: 2023
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DOI: 10.1080/03610926.2022.2045022

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