SUMS OF HAZARD FUNCTIONS OF EXPONENTIAL MIXTURES AND ASSOCIATED CONVOLUTIONS OF MIXED POISSON DISTRIBUTIONS

Abstract

A Sum of hazard functions of exponential mixtures characterizes a convolution of in nitely divisible mixed Poisson distributions which is also a convolution of compound Poisson distri- butions. For each sum of two special cases of Hofmann hazard function, the following have been ob- tained: the probability generating function (pgf) of the convolution of the mixed Poisson distri- butions. the pgf of the independent and identically distributed (iid) random variables for the convolution of the compound Poisson distributions. the recursive form of the convolution of the compound Poisson distribution. We also wish to nd out whether Panjer's recursive model holds for all cases.

Key words: convolutions, exponential mixtures, mixed Poisson distribution, Hofmann hazard functions, characterization, compound Poisson distribution, Panjer's recursive model, Laplace transform