Rare Event Simulation For Heavy-tailed Distributions Via Cross-entropy

Rare event refers to the event with extremely small probability, such as earthquakes, bankruptcy of insurance, system failure and so on. Rare event could cause mass destruction once it happened. Estimating the probability of rare event hasrealistic significance. Importance sampling is an effectivetechnique for rare event simulation, which overcomes the weakness of Monte Carlo simulation. The heavy-tailed distributionfamily is widely applied to the area of risk estimation. Minimum cross-entropy method is aneffective way for estimating the optimal importance sampling parameters, which minimizes the variance via minimizing the cross-entropy distance between the IS distribution and the zero-variance distribution. In this paper, we applied two minimum cross-entropy methods, the Multi-level Cross-entropy Method and the MLE Cross-entropy Method, to simulate the tail probabilities of two heavy-tailed distributions, the Lognormal distribution and the Log Gamma distribution, and the numerical results show that the Minimum Cross-entropy Method is superior to the Monte Carlo Simulation.