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Improved algorithms for rare event simulation with heavy tails

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen  and
Dirk P. Kroese
Søren Asmussen*
Affiliation:
University of Aarhus
Dirk P. Kroese*
Affiliation:
The University of Queensland
*
∗ Postal address: Department of Mathematical Sciences, Faculty of Science, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark. Email address: asmus@imf.au.dk
∗∗ Postal address: Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: kroese@maths.uq.edu.au
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Abstract

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The estimation of P(S n >u) by simulation, where S n is the sum of independent, identically distributed random varibles Y 1 ,…,Y n , is of importance in many applications. We propose two simulation estimators based upon the identity P(S n >u)=nP(S n >u, M n =Y n ), where M n =max(Y 1 ,…,Y n ). One estimator uses importance sampling (for Y n only), and the other uses conditional Monte Carlo conditioning upon Y 1 ,…,Y n−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

Keywords

Bounded relative errorcomplexityconditional Monte Carlo conditioningcontrol variatelogarithmic efficiencyM/G/1 queuePollaczek-Khinchin formularare eventregular variationstratificationsubexponential distributionWeibull distribution

MSC classification

Primary: 65C60: Computational problems in statistics
Secondary: 60K25: Queueing theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 545 - 558
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Partially supported by MaPhySto, the Danish National Research Foundation Network in Mathematical Physics and Stochastics, funded by the Danish National Research Foundation.

Supported by the Australian Research Council, grant number DP0558957.

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