Efficiency of Estimating Smooth Functions of Parameters in Rare-Event Contexts

Consider an estimand that is a known smooth function of an unknown vector of parameters, such as means, probabilities, quantiles, etc. Some but not necessarily all of the parameters relate to rare events, which typically make them difficult to estimate with naive Monte Carlo, so variance-reduction techniques must be applied to obtain efficient estimators of those parameters. We develop conditions ensuring that the overall estimand can be efficiently estimated, providing both sufficient and necessary-and-sufficient conditions. The paper illustrates the applicability of the theory through many examples and numerical results.