Recursive Formulas for Compound Phase Distributions: Univariate and Bivariate Cases

We first present a simple matrix-based recursive formula for calculating the distribution function of compound phase-type random variables. Then we extend the results to the case when the number of claims follows a bivariate matrix negative binomial (BMNB) distribution detailed herein. Further, extending the results in Hipp(2006), we provide speedy recursive formulas for both the univariate and the bivariate models when the claim sizes follow discrete phase-type distributions. Numerical examples are provided.

Keywords: compound phase distributions, recursive formulas, random variables.