Some Matrix-Variate Models Applicable in Different Areas

Abstract

Matrix-variate Gaussian-type or Wishart-type distributions in the real domain are widely used in the literature. When the exponential trace has an arbitrary power and when the factors involving a determinant and a trace enter into the model or a matrix-variate gamma-type or Wishart-type model with an exponential trace having an arbitrary power, they are extremely difficult to handle. One such model with factors involving a trace and a determinant and the exponential trace having an arbitrary power, in the real domain, is known in the literature as the Kotz model. No explicit evaluation of the normalizing constant in the Kotz model seems to be available. The normalizing constant that is widely used in the literature, is interpreted as the normalizing constant in the general model, and that is referred to as a Kotz model does not seem to be correct. One of the main contributions in this paper is the introduction of matrix-variate distributions in the real and complex domains belonging to the Gaussian-type, gamma-type, and type 1 and type 2 beta-types, or Mathai’s pathway family, when the exponential trace has an arbitrary power and explicit evaluations of the normalizing constants therein. All of these models are believed to be new. Another new contribution is the logistic-based extensions of the models in the real and complex domains, with the exponential trace having an arbitrary exponent and connecting to extended zeta functions introduced by this author recently. The techniques and steps used at various stages in this paper will be highly useful for people working in multivariate statistical analysis, as well as for people applying such models in engineering problems, communication theory, quantum physics, and related areas, apart from statistical applications.