Density Estimators of the Cumulative Reward Up to a Hitting Time to a Rarely Visited Set of a Regenerative System

For a regenerative process, we propose various estimators of the density function of the cumulative reward up to hitting a rarely visited set of states. The approaches exploit existing weak-convergence results for the hitting-time distribution, and we apply simulation (often with previously developed importance samplers for estimating the mean) to estimate parameters of the limiting distribution. We also combine these ideas with kernel methods. Numerical results from simulation experiments show the effectiveness of the estimators.