Minimum Cross-entropy Methods for Rare-event Simulation

We apply the minimum cross-entropy method (MinXEnt) for estimating rare-event probabilities for the sum of i.i.d. random variables. MinXEnt is an analogy of the MaXimum Entropy Principle in the sense that the objective is to minimize a relative (or cross) entropy of a target density h from an unknown density f under suitable constraints. The main idea is to use the solution to this optimization program as the simulation density in importance sampling. We shall see that some eXisting importance sampling methods can be cast in a MinXEnt program, such as the large deviations approach for light tails and the hazard rate twisting for heavy tails. As an eXtension, we shall consider a correlated version of this hazard rate twisted solution which gives better simulation results. The sample generation is based on a Gibbs sampler algorithm.