Properties of Matrix Variate Beta Type 3 Distribution

Arjun K Gupta,Daya K Nagar

doi:10.1155/2009/308518

Abstract

We study several properties of matrix variate beta type 3 distribution. We also derive probability density functions of the product of two independent random matrices when one of them is beta type 3. These densities are expressed in terms of Appell's first hypergeometric functionF1and Humbert's confluent hypergeometric functionΦ1of matrix arguments. Further, a bimatrix variate generalization of the beta type 3 distribution is also defined and studied.

Highlights

The beta families of distributions are defined by the density functions uα−1 1−u β−1 0 < u < 1,B α, β vα−1 1 v − α β, v > 0, respectively, where α > 0, β > 0, and ΓαΓ β Γα βThe beta type 1 and beta type 2 are very flexible distributions for positive random variables and have wide applications in statistical analysis, for example, see Johnson et al 1 .International Journal of Mathematics and Mathematical SciencesRecently, Cardeno et al 2 have defined and studied family of beta type 3 distributions
We study several properties of matrix variate beta type 3 distribution
The matrix variate beta type 3 distribution has been defined, and some of its properties have been studied by Gupta and Nagar 4

Summary

Introduction

The beta type 1 and beta type 2 are very flexible distributions for positive random variables and have wide applications in statistical analysis, for example, see Johnson et al 1. Cardeno et al 2 have defined and studied family of beta type 3 distributions. A random variable w is said to follow a beta type 3 distribution if its density function is given by. The matrix variate generalizations of 1.1 and 1.2 have been studied extensively in the literature, for example, see Gupta and Nagar 3. The matrix variate beta type 3 distribution has been defined, and some of its properties have been studied by Gupta and Nagar 4. We study several properties of matrix variate beta type 3 distribution. We derive probability density functions of the product of two independent random matrices when one of them is beta type 3. We define bimatrix beta type 3 distribution and study some of its properties

Some Known Results and Definitions

Hypergeometric Functions of Two Matrices

Properties

Distributions of Random Quadratic Forms

Bimatrix Beta Type 3 Distribution