Link Between Mixed Poisson Distributions and Exponential Mixtures

The hazard function of an exponential mixture characterizes an infinitely divisible mixed Poisson distribution which is also a compound Poisson distribution. A sum of hazard functions of exponential mixtures characterizes a convolution of infinitely divisible mixed Poisson distributions which is also a convolution of compound Poisson distributions. Laplace transforms of sums of independent continuous random variables has been obtained as an alternative method of obtaining hazard functions of exponential mixtures. Using the relationship between hazard functions and mean excess function, (also known as mean residual lifetime) of a distribution, the hazard function of Benktander II distribution is obtained as a sum of hazard functions of a Pareto and exponential-Hougaard distributions. Given the hazard function of an exponential mixture, the probability generating functions (pgf) of the compound Poisson distribution and its independent and identically distributed (iid) random variables are derived. The recursive form of the distribution of the iid random variables for the Hofmann distribution follows Panjer's model. Hofmann hazard function has been discussed and re-parameterized.