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

Abstract

In this paper we use Importance Sampling to estimate tail probabilities for a finite sum of lognormal distributions. We use a defensive mixture, and develop a method of choosing the parameters via the EM algorithm; we also consider the technique which assumes the importance sampling density to belong to the same parametric family of the random variables to be summed. In both cases, the instrumental density is found by minimizing Cross-Entropy. A comparison based on several simulation experiments shows that the defensive mixture has the best performance. Finally, we study the Poisson-lognormal compound distribution framework and present a real-data application.

Keywords:

• Compound distributions
• Cross-entropy
• Defensive mixtures
• Importance sampling
• Tail probability

Mathematics Subject Classification:

• 62F10
• 62P05

Acknowledgments

I would like to thank Leah Flury Wetherill and three anonymous referees for valuable comments on an earlier version of the article.