HEAVY TAILS, IMPORTANCE SAMPLING AND CROSS–ENTROPY

ABSTRACT We consider the problem of estimating ℙ(Y 1 + … + Y n > x) by importance sampling when the Y i are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a tool for choosing good parameters in the importance sampling distribution; in doing sso, we use the asymptotic description that given ℙ(Y 1 + … + Y n > x), n − 1 of the Y i have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/1 and GI/G/1 queues are also discussed.

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