Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications

Daya K Nagar,Sergio Alexander Gómez-Noguera,Arjun K Gupta

doi:10.17230/ingciencia.12.24.3

Daya K Nagar, Sergio Alexander Gómez-Noguera + Show 1 more

Open Access

https://doi.org/10.17230/ingciencia.12.24.3

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Abstract

In this article, we define and study generalized forms of extended matrixvariate gamma and beta functions. By using a number of results from ma-trix algebra, special functions of matrix arguments and zonal polynomialswe derive a number of properties of these newly defined functions. We alsogive some applications of these functions to statistical distribution theory.

Highlights

The gamma function was first introduced by Leonard Euler in 1729, as the limit of a discrete expression and later as an absolutely convergent improper integral, namely, ∞Γ(a) = ta−1 exp(−t) dt, Re(a) > 0. (1)The gamma function has many beautiful properties and has been used in almost all the branches of science and engineering.One year later, Euler introduced the beta function defined for a pair of complex numbers a and b with positive real parts, through the integralB(a, b) = ta−1(1 − t)b−1 dt, Re(a) > 0, Re(b) > 0. (2)The beta function has many properties, including symmetry, B(a,b) = B(b,a), and its relationship to the gamma function, Γ(a)Γ(b) B(a, b) = Γ(a + b)
It is clear that if Σ = 0, for Re(a) > (m−1)/2, the extended matrix variate gamma function reduces to the multivariate gamma function Γm(a)
A matrix variate generalization of the generalized extended gamma function can be defined in the following way: Definition 3.1

Summary

Introduction

The gamma function was first introduced by Leonard Euler in 1729, as the limit of a discrete expression and later as an absolutely convergent improper integral, namely,. The domains of gamma and beta functions have been extended to the whole complex plane by introducing in the integrands of (1) and (2), the factors exp (−σ/t) and exp [−σ/t(1 − t)], respectively, where Re(σ) > 0. Generalizations of the extended gamma and extended beta functions defined by (5) and (6), respectively, to the matrix case have not been defined and studied.

Some known definitions and results

Generalized extended matrix variate Gamma function

Generalized extended matrix variate Beta function

Results involving zonal polynomials

Application to multivariate statistics

Generalized extended matrix variate Beta distribution