On Wishart Ratios with Dependent Structure
Andriëtte Bekker¹, M. Arashi², J.J.J. Roux¹
¹Statistics, University of Pretoria, Pretoria, Gauteng, South Africa; ²Faculty of Mathematics, Shahrood University of Technology, Shahrood, Islamic Republic of Iran

In multivariate statistics, the problems of interest are random covariance matrices (known as Wishart matrices) and ratios of Wishart matrices that arise in multivariate analysis of variance (MANOVA) (see Muirhead, 1982). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart random variates (see Bekker, Roux, Ehlers and Arashi, 2010; Gupta and Nagar, 2009; Diaz-Garzia and Gutiérrez-Jámez, 2010)). In this presentation, we add a further independent Wishart random variate to the “ratios” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. Some interesting characteristics are explored. Lastly, we focus on the bivariate case of this distribution (is an extension of Jones' bivariate beta) with emphasis on stress-strength model.

Bekker, A., Roux, J.J.J., Ehlers, R. & Arashi, M. (2010). Bimatrix variate beta type IV distribution: relation to Wilks' statistic and bimatrix variate Kummer-beta type IV distribution. Communications in Statistics- Theory and Methods (in production).

Diaz-Garzia, A & Gutiérrez-Jámez, R. (2010). Bimatrix variate generalised beta distributions. South African Statist. J. 44, 193-208.

Gupta, A.K. & Nagar, D.K. (2009). Matrix variate generalization of a bivariate beta type I distribution. Journal of Statistics and Management Systems, 12 (5), 873-885.

Jones, M.C. (2001). Multivariate t and the beta distributions associated with the multivariate F distribution, Metrika, 54, 215-231.

Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. New York: John Wiley & Sons.

Keywords: bimatrix variate beta type IV distribution; Ratios; Wishart distribution; Extension of Jones' bivariate beta model

Biography: Professor Andriëtte Bekker's career spans 30 years in an academic university environment and involved appointments at the University of Johannesburg and University of South Africa. She currently lectures at the University of Pretoria, South Africa. Her main research interest is distribution theory. She is married and has two children.