Importance Sampling for Sums of Lognormal Distributions, with Applications to Operational Risk

In this paper we estimate tail probabilities for the sum of Lognormal distributions. We propose to use a defensive mixture, and develop a method of finding the optimal density via the EM algorithm; we also consider the technique which assumes the importance sampling density to belong to the same parametric family of the distribution of the random variables to be summed. Optimality is defined in terms of minimal Cross-Entropy. Several simulation experiments show that the defensive mixture has the best performance. Finally, we study the compound distribution framework, and present a real-data application concerning the Poisson-Lognormal compound distribution.