Efficiency of importance sampling estimators

This paper deals with the efficiency of importance sampling simulations for estimating probabilities of rare events, in particular, when these probabilities become asymptotically small. We investigate for importance sampling estimators, the properties of bounded relative error (BRE), bounded normal approximation (BNA) and asymptotic optimality (AO). We represent the criteria in a unified manner and we show formally that AO is a strictly weaker criterion than the others. Besides, BRE is strictly weaker than BNA thus yielding a hierarchy of efficiency criteria. Furthermore, we develop conditions for AO in cases, where the relative error is unbounded. These constitute a characterization of asymptotically optimal importance sampling estimators involving the ratio of the orders of magnitude of the relative error and the unknown quantity to be estimated, typically a rare event probability.