The compounding method for finding bivariate noncentral distributions

Abstract

The univariate and bivariate central chi-square- and F distributions have received a decent amount of attention in the literature during the past few decades; the noncentral counterparts of these distributions have been much less present. This study 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 used to a limited extent in the field of distribution theory to obtain univariate noncentral distributions; this study extends some results in literature to the corresponding bivariate setting. The process which is followed to obtain such bivariate noncentral distributions is systematically described and motivated. Some distributions of composites (univariate functions of the dependent components of the bivariate distributions) are derived and studied, in particular the product, ratio, and proportion. The benefit of introducing these bivariate noncentral distributions and their respective composites is demonstrated by graphical representations of their probability density functions. Furthermore, an example of possible application is given and discussed to illustrate the versatility of the proposed models.