Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-02T05:03:14.451Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison of cross-entropy and variance minimization strategies

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Joshua C. C. Chan ,
Peter W. Glynn  and
Dirk P. Kroese
Joshua C. C. Chan
Affiliation:
Australian National University, Research School of Economics, Australian National University, Canberra, ACT 0200, Australia
Peter W. Glynn
Affiliation:
Stanford University, Department of Management Science and Engineering, Huang Engineering Center, Stanford University, Stanford University, 475 Via Ortega, Stanford, CA 94305-4121, USA
Dirk P. Kroese
Affiliation:
University of Queensland, Department of Mathematics, University of Queensland, St Lucia, Brisbane, QLD 4072, Australia. Email address: kroese@maths.uq.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.

Keywords

Variance minimizationcross entropyimportance samplingrare-event simulationlikelihood ratio degeneracy

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 65C60: Computational problems in statistics
Type
Part 4. Simulation
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 183 - 194
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S. and Glynn, P. W., (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
[2] Asmussen, S. and Kroese, D. P., (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558.Google Scholar
[3] Asmussen, S., Binswanger, K. and Ho jgaard, B., (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322.Google Scholar
[4] Asmussen, S., Kroese, D. P. and Rubinstein, R. Y., (2005). Heavy tails, importance sampling and cross-entropy. Stoch. Models 21, 57–76.CrossRefGoogle Scholar
[5] Blanchet, J. and Glynn, P., (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378.CrossRefGoogle Scholar
[6] Blanchet, J. and Li, C., (2011). Efficient rare event simulation for heavy-tailed compound sums. To appear in ACM Trans. Model. Comput. Simul. CrossRefGoogle Scholar
[7] Casella, G. and Berger, R. L., (2001). Statistical Inference, 2nd edn. Duxbury Press.Google Scholar
[8] Chan, J. C. C. and Kroese, D. P., (2011). Improved cross-entropy method for estimation. Submitted.Google Scholar
[9] Chan, J. C. C. and Kroese, D. P., (2009). Rare-event probability estimation with conditional Monte Carlo. Ann. Operat. Res., 19pp.Google Scholar
[10] 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
[11] Kroese, D. P., (2011). The cross-entropy method. In Wiley Encyclopedia of Operations Research and Management Science.Google Scholar
[12] Kroese, D. P. and Rubinstein, R. Y., (2004). The transform likelihood ratio method for rare event simulation with heavy tails. Queueing Systems 46, 317–351.CrossRefGoogle Scholar
[13] L'Ecuyer, P., Blanchet, J. H., Tuffin, B. and Glynn, P. W., (2010). Asymptotic robustness of estimators in rare-event simulation. ACM Trans. Model. Comput. Simul. 20, 1–41.CrossRefGoogle Scholar
[14] Rubinstein, R. Y., (2009). The Gibbs cloner for combinatorial optimization, counting and sampling. Methodology Comput. Appl. Prob. 11, 491–549.Google Scholar
[15] Rubinstein, R. Y. and Glynn, P. W., (2009). How to deal with the curse of dimensionality of likelihood ratios in Monte Carlo simulation. Stoch. Models 25, 547–568.Google Scholar
[16] Rubinstein, R. Y. and Kroese, D. P., (2004). The Cross-Entropy Method. Springer, New York.Google Scholar
[17] Rubinstein, R. Y. and Kroese, D. P., (2007). Simulation and the Monte Carlo Method, 2nd edn. John Wiley, New York.Google Scholar