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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(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−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
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.

References

Abate, J., Choudhury, G. L. and Whitt, W. (1994). Waiting-time tail probabilities in queues with long-tailed service-time distributions. Queueing Systems 16, 311–338.Google Scholar
Adler, R. J., Feldman, R. and Taqqu, M. S. (eds) (1998). A Practical Guide to Heavy Tails. Birkhäuser, Boston, MA.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Binswanger, K. (1997). Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27, 297–318.CrossRefGoogle Scholar
Asmussen, S. and Rubinstein, R. Y. (1995). Steady state rare events simulation in queueing models and its complexity properties. In Advances in Queueing. Theory, Methods, and Open Problems, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 429–466.Google Scholar
Asmussen, S., Binswanger, K. and Højgaard, B. (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322.Google Scholar
Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at subexponential times, with queueing applications. Stoch. Process. Appl. 79, 265–286.CrossRefGoogle Scholar
Asmussen, S., Kroese, D. P. and Rubinstein, R. Y. (2005). Heavy tails, importance sampling and cross-entropy. Stoch. Models 21, 57–76.Google Scholar
Baltrunas, A., Daley, D. J. and Klüppelberg, C. (2004). Tail behaviour of the busy period of a GI/G/1 queue with subexponential service times. Stoch. Process. Appl. 111, 237–258.Google Scholar
Cruz, M. G. (2002). Modeling, Measuring and Hedging Operational Risk. John Wiley, Chichester.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Frachot, A., Moudoulaud, O. and Roncalli, T. (2004). Loss distribution approach in practice. In The Basel Handbook. A Guide for Financial Practitioners, ed. Ong, M. K., Risk Books, London.Google Scholar
Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.Google Scholar
Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. ACM Trans. Model. Comput. Simul. 5, 43–85.Google Scholar
Juneja, S. and Shahabuddin, P. (2002). Simulating heavy tailed processes using delayed hazard rate twisting. ACM Trans. Model. Comput. Simul. 12, 94–118.Google Scholar
Mikosch, T. and Nagaev, A. V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81–110.Google Scholar
Rubinstein, R. Y. and Kroese, D. P. (2004). The Cross-Entropy Method. A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. Springer, New York.Google Scholar
Sigman, K. (1999). Appendix: a primer on heavy-tailed distributions. Queueing Systems 33, 261–275.Google Scholar