On distributions of covariance structures

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 $u=X\prime AY$ where X and Y are $p\times 1$ and $q\times 1$ 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 ${(1-{\lambda }^2{t}^2)}^{-\alpha },\lambda >0,\alpha >0,-\frac{1}{\lambda }<\mathfrak{R}(t)<\frac{1}{\lambda }$ where t is the mgf parameter, $\mathfrak{R}(·)$ means the real part of $(·),$ and λ and α are real scalar parameters. Explicit evaluation of the density of u is considered when α is a positive integer as well as for a general α. It is shown that the exact densities can be written as linear functions of double gamma densities and double exponential or Laplace densities when α is a positive integer. For the general value of α, it is shown that the exact density can be written in terms of double Mittag-Leffler or a double confluent hypergeometric function.

Keywords:

• Covariance structures
• bilinear forms
• moment generating function
• Laplace transforms
• double gamma
• double exponential densities
• double hypergeometric functions
• generalized partial fractions

2020 Mathematics subject classification numbers:

• 62H10
• 15A63
• 15B52
• 33C15
• 44A10

Acknowledgments

The authors would like to thank the referees for their valuable comments, which enabled the authors to improve the presentation of the material in the paper. The authors declare no conflict of interest. This research is not supported by any grant from any granting agency.