Jacobus J.J. Roux

Bimatrix Variate Beta Type IV Distribution: Relation to Wilks's Statistic and Bimatrix Variate Kummer-Beta Type IV Distribution

In this article, the bimatrix variate beta Type IV distribution is derived from independent Wishart distributed matrix variables. We explore specific properties of this distribution which is then used to derive the exact expressions of the densities of the product and ratio of two dependent Wilkss statistics and to define the bimatrix Kummer-beta Type IV distribution.

Subjective Bayesian Analysis of the Elliptical Model

For the multivariate elliptical model subjective Bayesian estimators of the location vector and some functions of the characteristic matrix with the normal-inverse Wishart and the normal-Wishart as prior, respectively, are derived. Fang and Li (1999) considered the elliptical model for Bayesian analysis for an objective prior structure. In addition, the newly developed results are applied to the multivariate normal- and t-distribution. A performance study is done to evaluate the normal-gamma and normal-inverse gamma distributions as suitable priors. A practical application for the posterior distributions of the multivariate t-distribution is included by means of Gibbs sampling and a Metropolis-Hastings algorithm.

On the real representation of quaternion random variables

For the first time, the matrix-variate quaternion normal and quaternion Wishart distributions are derived from first principles, that is, from their real counterparts, exposing the relations between their respective densities and characteristic functions. Applications of this theory in hypothesis testing are presented, and the density function of Wilks’ statistic is derived for quaternion Wishart matrices.

Integral Representation of Quaternion Elliptical Density and its Applications

In this article, an integral representation for the density of a matrix variate quaternion elliptical distribution is proposed. To this end, a weight function is used, based on the inverse Laplace transform of a function of a Hermitian quaternion matrix. Examples of well-known members of the family of quaternion elliptical distributions are given as well as their respective weight functions. It is shown that under some conditions, the proposed formula can be applied for the scale mixture of quaternion normal models. Applications of the proposed method are also given.

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.