New efficient estimators in rare event simulation with heavy tails

This paper is concerned with the efficient simulation of P(Sn>s)P(Sn>s) in situations where ss is large and SnSn is the sum of nn i.i.d. heavy-tailed random variables X1,…,XnX1,…,Xn. The most efficient and simplest estimators introduced in the rare event simulation literature are those proposed by Asmussen and Kroese (2006) and Asmussen and Kortschak (2012). Although the main techniques for facing the rare event problem are importance sampling and splitting, the estimators of Asmussen, Kortschak and Kroese combine exchangeability arguments with conditional Monte-Carlo to construct estimators whose relative errors go to 0 as s→∞s→∞. In this paper, we decompose P(Sn>s)P(Sn>s) as the sum of P(Mn>s)P(Mn>s) and P(Sn>s,Mns,Mns)P(Mn>s) is known in closed form and is asymptotically equivalent to P(Sn>s)P(Sn>s). We construct new efficient estimators of P(Sn>s,Mns,Mn