Importance sampling (IS) is the primary technique for constructing reliable estimators in the context of rare-event simulation. The asymptotic robustness of IS estimators is often qualified by properties such as bounded relative error (BRE) and asymptotic optimality (AO). These properties guarantee that the estimator’s relative error remains bounded (or does not increase too fast) when the rare events becomes rarer. Other recently introduced characterizations of IS estimators are bounded normal approximation (BNA), bounded relative efficiency (BREff), and asymptotic good estimation of mean and variance.

In this paper we introduce three additional property named bounded relative error of empirical variance (BREEV), bounded relative efficiency of empirical variance (BREffEV), and asymptotic optimality of empirical variance (AOEV), which state that the empirical variance has itself the BRE, BREff and AO property, respectively, as an estimator of the true variance. We then study the hierarchy between all these different characterizations for a model of highly-reliable Markovian systems (HRMS) where the goal is to estimate the failure probability of the system. In this setting, we show that BRE, BREff and AO are equivalent, that BREffEV, BREEV and AOEV are also equivalent, and that these two properties are strictly stronger than all other properties just mentioned. We also obtain a necessary and sufficient condition for BREEV in terms of quantities that can be readily verified from the parameters of the model.