On distributions of covariance structures

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 px1 and qx1 real vectors and A is a constant p x 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-lambda(2)t(2))(-alpha),lambda > 0,alpha > 0,-1/lambda<1/lambda where t is the mgf parameter, R(center dot) means the real part of (center dot), and lambda and alpha are real scalar parameters. Explicit evaluation of the density of u is considered when alpha is a positive integer as well as for a general alpha. It is shown that the exact densities can be written as linear functions of double gamma densities and double exponential or Laplace densities when alpha is a positive integer. For the general value of alpha, it is shown that the exact density can be written in terms of double Mittag-Leffler or a double confluent hypergeometric function.

Automated summary

This paper discusses the distribution of covariance structures in real-world scalar/vector/matrix variables, using bilinear forms and Laplace transforms or moment generating functions. The results show that the density function can be expressed as a linear function of double gamma densities or double exponential or Laplace densities when alpha is a positive integer, and in terms of double Mittag-Leffler functions or double confluent hypergeometric functions for the general value of alpha.