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On error rates in rare event simulation with heavy tails

By Søren Asmussen and Dominik Kortschak
Thiele Research Reports
No. 01, May 2012
For estimating $\mathbb{B}(S_n > x)$ by simulation where $S_k=Y_1+\cdots+Y_k$ with $Y_1,\ldots,Y_n$ are heavy-tailed with distribution $F$, (Asmussen and Kroese 2006) suggested the estimator $n\,\overline F\bigl(M_{n-1}\vee(x-S_{n-1})\bigr)$ where $M_k=\max(Y_1,\ldots,Y_k)$. The estimator has shown to perform excellently in practice and has also nice theoretical properties. In particular, (Hartinger and Kortschak 2009) showed that the relative error goes to 0 as $x\to\infty$. We identify here the exact rate of decay and propose some related estimators with even faster rates.
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