Let $Y_1,\ldots,Y_n$ be i.i.d. subexponential and $S_n=Y_1+\cdots+Y_n$. Asmussen and Kroese (2006) suggested a simulation estimator for evaluating $\mathbb{P}(S_n>x)$, combining an exchangeability argument with conditional Monte Carlo. The estimator was later shown by Hartinger & Kortschak (2009) to have vanishing relative error. For the Weibull and related cases, we calculate the exact error rate and suggest improved estimators. These improvements can be seen as control variate estimators, but are rather motivated by second order subexponential theory which is also at the core of the technical proofs.

Keywords:Complexity, conditional Monte Carlo, control variates, lognormal distribution, M/G/1 queue, Pollaczeck-Khinchine formula, rare event, regular variation, ruin theory, second order subexponentiality, subexponential distribution, vanishing relative error, Weibull distribution