Estimating tail probabilities of random sums of infinite mixtures of phase-type distributions

We consider the problem of estimating tail probabilities of random sums of infinite mixtures of phase-type (IMPH) distributions—a class of distributions corresponding to random variables which can be represented as a product of an arbitrary random variable with a classical phase-type distribution. Our motivation arises from applications in risk and queueing problems. Classical rare-event simulation algorithms cannot be implemented in this setting because these typically rely on the availability of the CDF or the MGF, but these are difficult to compute or not even available for the class of IMPH distributions. In this paper, we address these issues and propose alternative simulation methods for estimating tail probabilities of random sums of IMPH distributions; our algorithms combine importance sampling and conditional Monte Carlo methods. The empirical performance of each method suggested is explored via numerical experimentation.