Improving the Asmusse}--Kroese-type simulation estimators

Abstract

The Asmussen--Kroese Monte Carlo estimators of P(S_n > u) and P(S_N > u) are known to work well in rare event settings, where S_N is the sum of independent, identically distributed heavy-tailed random variables X₁,...,X_N and N is a nonnegative, integer-valued random variable independent of the X_i. In this paper we show how to improve the Asmussen--Kroese estimators of both probabilities when the X_i are nonnegative. We also apply our ideas to estimate the quantity E[(S_N-u)₊].