The Compounding Method for Finding Bivariate Noncentral Chi-Square and F Distributions

• Activity Number: 602
• Type: Contributed
• Date/Time: Thursday, August 7, 2014 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Johannes Ferreira*+ and Andriette Bekker and Mohammad Arashi
• Companies: University of Pretoria and University of Pretoria and Shahrood University
• Keywords: noncentral bivariate chi-square ; noncentral bivariate F ; noncentral ; Poisson ; compounding ; composites

Abstract

The univariate and bivariate central chi-square and F distributions have received a decent amount of attention in the literature over the past few decades; the noncentral counterparts of these distributions have been much less present. This paper enriches the existing literature by proposing bivariate noncentral chi-square and F distributions via the employment of the compounding method with Poisson probabilities. This method has been suggested by Marshall and Olkin (1990, Topics in Statistical Dependence, 371-393) and more recently by Yunus and Khan (2011, Applied Mathematics and Computation 217, 6237-6247). Some distributions of composites (univariate functions of the bivariate components) are derived and studied. The benefit of introducing these distributions and their respective composites (such as the product and ratio of components) will be demonstrated by graphical representations of their density functions. An example using rainfall data is given to illustrate the versatility of the proposed models.