skip to main content
10.5555/3042094.3042152acmconferencesArticle/Chapter ViewAbstractPublication PageswscConference Proceedingsconference-collections
research-article

An M-estimator for rare-event probability estimation

Published: 11 December 2016 Publication History

Abstract

We describe a maximum-likelihood type estimator, or M-estimator, for Monte Carlo estimation of rare-event probabilities. In this method, we first sample from the zero-variance measure using Markov Chain Monte Carlo (MCMC), and then given the simulated data, we compute a maximum-likelihood-type estimator. We show that the resulting M-estimator is consistent, and that it subsumes as a special case the well-known fixed-effort splitting estimator. We give a numerical example of estimating accurately the tail distribution of the sum of log-normal random variables under a Gaussian copula. The numerical results suggests that for this example the method is competitive.

References

[1]
Asmussen, S., J. Blanchet, S. Juneja, and L. Rojas-Nandayapa. 2011. "Efficient Simulation of Tail Probabilities of Sums of Correlated Lognormals". Annals of Operations Research 189 (1): 5--23.
[2]
Asmussen, S., and P. W. Glynn. 2007. Stochastic Simulation: Algorithms and Analysis. New York: Springer-Verlag.
[3]
Asmussen, S., J. L. Jensen, and L. Rojas-Nandayapa. 2014. "Exponential Family Techniques for the Lognormal Left Tail". arXiv preprint arXiv:1403.4689.
[4]
Botev, Z. I. 2009. "Splitting Methods for Efficient Combinatorial Counting and Rare-Event Probability Estimation". PhD thesis: The University of Queensland Library.
[5]
Botev, Z. I., and D. P. Kroese. 2008. "An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting". Methodology and Computing in Applied Probability 10 (4): 471--505.
[6]
Botev, Z. I., and D. P. Kroese. 2012. "Efficient Monte Carlo Simulation via the Generalized Splitting method". Statistics and Computing 22 (1): 1--16.
[7]
Botev, Z. I., P. L'Ecuyer, and B. Tuffin. 2011. "An Importance Sampling Method Based on a One-step Look-ahead Density from a Markov Chain". In Proceedings of the 2011 Winter Simulation Conference, edited by S. Jain, R. R. Creasley, J. Himmelspach, K. P. White, and M. Fu, 528--539. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc.
[8]
Botev, Z. I., P. L'Ecuyer, and B. Tuffin. 2013. "Markov Chain Importance Sampling with Applications to Rare-event Probability Estimation". Statistics and Computing 23 (2): 271--285.
[9]
Botev, Z. I., M. Mandjes, and A. Ridder. 2015. "Tail Distribution of the Maximum of Correlated Gaussian Random Variables". In Proceedings of the 2015 Winter Simulation Conference, edited by L. Yilmaz, W. K. V. Chan, I. Moon, T. M. K. Roeder, C. Macal, and M. D. Rossetti, 633--642. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc.
[10]
Botev, Z. I., A. Ridder, and L. Rojas-Nandayapa. 2016. "Semiparametric Cross Entropy For Rare-Event Simulation". Journal of Applied Probability 53 (3).
[11]
Doss, H., and A. Tan. 2014. "Estimates and Standard Errors for Ratios of Normalizing Constants from Multiple Markov Chains via Regeneration". Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76 (4): 683--712.
[12]
Gill, R. D., Y. Vardi, and J. A. Wellner. 1988. "Large Sample Theory of Empirical Distributions in Biased Sampling Models". The Annals of Statistics 16 (3): 1069--1112.
[13]
Gudmundsson, T., and H. Hult. 2014. "Markov Chain Monte Carlo for Computing Rare-event Probabilities for a Heavy-tailed Random Walk". Journal of Applied Probability 51 (2): 359--376.
[14]
Huang, A., and Z. I. Botev. 2013. "Rare-event Probability Estimation via Empirical Likelihood Maximization". arXiv preprint arXiv:1312.3027.
[15]
Kong, A., P. McCullagh, X.-L. Meng, D. Nicolae, and Z. Tan. 2003. "A Theory of Statistical Models for Monte Carlo Integration". Journal of the Royal Statistical Society, Series B 65 (3): 585--618.
[16]
Kortschak, D., and E. Hashorva. 2013. "Efficient Simulation of Tail Probabilities for Sums of Log-elliptical Risks". Journal of Computational and Applied Mathematics 247:53--67.
[17]
Kroese, D. P., T. Taimre, and Z. I. Botev. 2011. Handbook of Monte Carlo Methods, Volume 706. Wiley.
[18]
Laub, P. J., S. Asmussen, J. L. Jensen, and L. Rojas-Nandayapa. 2015. "Approximating the Laplace Transform of the Sum of Dependent Lognormals". arXiv preprint arXiv:1507.03750.
[19]
Vardi, Y. 1985. "Empirical Distributions in Selection Bias Models". The Annals of Statistics 13 (1): 178--203.

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
WSC '16: Proceedings of the 2016 Winter Simulation Conference
December 2016
3974 pages
ISBN:9781509044849

Sponsors

In-Cooperation

  • SAS
  • AnyLogic: The AnyLogic Company
  • Palgrave: Palgrave Macmillan
  • FlexSim: FlexSim Software Products, Inc.
  • ASA: American Statistical Association
  • IEEE/SMC: Institute of Electrical and Electronics Engineers: Systems, Man, and Cybernetics Society
  • Simio: Simio LLC
  • ODU: Old Dominion University
  • ASIM: Arbeitsgemeinschaft Simulation
  • ExtendSim: ExtendSim
  • NIST: National Institute of Standards & Technology
  • Amazon Simulations: Amazon Simulations

Publisher

IEEE Press

Publication History

Published: 11 December 2016

Check for updates

Qualifiers

  • Research-article

Conference

WSC '16
Sponsor:
WSC '16: Winter Simulation Conference
December 11 - 14, 2016
Virginia, Arlington

Acceptance Rates

Overall Acceptance Rate 3,413 of 5,075 submissions, 67%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 35
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 13 Jan 2025

Other Metrics

Citations

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media