The Annals of Applied Probability

Counterexamples in importance sampling for large deviations probabilities

Paul Glasserman and Yashan Wang

Full-text: Open access

Abstract

A guiding principle in the efficient estimation of rare-event probabilities by Monte Carlo is that importance sampling based on the change of measure suggested by a large deviations analysis can reduce variance by many orders of magnitude. In a variety of settings, this approach has led to estimators that are optimal in an asymptotic sense. We give examples, however, in which importance sampling estimators based on a large deviations change of measure have provably poor performance. The estimators can have variance that decreases at a slower rate than a naive estimator, variance that increases with the rarity of the event, and even infinite variance. For each example, we provide an alternative estimator with provably efficient performance. A common feature of our examples is that they allow more than one way for a rare event to occur; our alternative estimators give explicit weight to lower probability paths neglected by leading-term asymptotics.

Article information

Source
Ann. Appl. Probab., Volume 7, Number 3 (1997), 731-746.

Dates
First available in Project Euclid: 16 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034801251

Digital Object Identifier
doi:10.1214/aoap/1034801251

Mathematical Reviews number (MathSciNet)
MR1459268

Zentralblatt MATH identifier
0892.60043

Subjects
Primary: 60F10: Large deviations 60J15 65C05: Monte Carlo methods

Keywords
Monte Carlo methods rare events random walks large deviations importance sampling simulation

Citation

Glasserman, Paul; Wang, Yashan. Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7 (1997), no. 3, 731--746. doi:10.1214/aoap/1034801251. https://projecteuclid.org/euclid.aoap/1034801251


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References

  • 1 ANANTHARAM, V., HEIDELBERGER, P. and TSOUCAS, P. 1990. Analy sis of rare events in continuous time Markov chains via time reversal and fluid approximations. Research Report RC-16280, IBM, Yorktown Heights, NY.
  • 2 ASMUSSEN, S. 1982. Conditioned limit theorems relating a random walk to its associate. Adv. in Appl. Probab. 14 143 170.
  • 3 ASMUSSEN, S. 1985. Conjugate processes and the simulation of ruin probabilities. Stochastic Process. Appl. 20 213 229.
  • 4 ASMUSSEN, S. 1987. Applied Probability and Queues. Wiley, Chichester.
  • 5 ASMUSSEN, S. and BINSWANGER, K. 1995. Simulation of ruin probabilities for subexponential claims. Working paper, Dept. Mathematical Statistics, Univ. Lund, Sweden.
  • 6 ASMUSSEN, S., RUBINSTEIN, R. and WANG, C. L. 1994. Regenerative rare event simulation via likelihood ratios. J. Appl. Probab. 31 797 815.
  • 7 BAZARAA, M. S. and SHETTY, C. M. 1979. Nonlinear Programming: Theory and Algorithms. Wiley, New York.
  • 8 BUCKLEW, J. A. 1990. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York.
  • 9 BUCKLEW, J. A., NEY, P. and SADOWSKY, J. S. 1990. Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains. J. Appl. Probab. 27 44 59.
  • 10 CHANG, C. S., HEIDELBERGER, P., JUNEJA, P. and SHAHABUDDIN, P. 1994. Effective bandwidth and fast simulation of ATM intree networks. Performance Evaluation 20 45 65.
  • 11 CHANG, C. S., HEIDELBERGER, P. and SHAHABUDDIN, P. 1995. Fast simulation of packet loss rates in a shared buffer communications switch. ACM Trans. Modeling Comp. Simulation 5 306 325.
  • 12 CHEN, J.-C., LU, D., SADOWSKY, J. S. and YAO, K. 1993. On importance sampling in digital communications. I: fundamentals. IEEE J. Selected Areas in Comm. 11 289 307.
  • 13 DEMBO, A. and ZEITOUNI, O. 1993. Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
  • 14 FRATER, M., LENNON, T. M. and ANDERSON, B. D. O. 1991. Optimally efficient estimation of the statistics of rare events in queuing networks. IEEE Trans. Automat. Control 36 1395 1405.
  • 15 FRESNEDO, R. D. 1989. Quick simulation of rare events in networks. In Proceedings of the Winter Simulation Conference 514 523. Society for Computer Simulation, San Diego.
  • 16 GLASSERMAN, P. and KOU, S. 1995. Analy sis of an importance sampling estimator for tandem queues. ACM Trans. Modeling Comp. Simulation 4 22 42.
  • 17 GLy NN, P. W. 1995. Large deviations for the infinite server queue in heavy traffic. IMA Vol. Math. Appl. 71 387 394.
  • 18 HEIDELBERGER, P. 1995. Fast simulation of rare events in queueing and reliability models. ACM Trans. Modeling Comp. Simulation 4 43 85.
  • 19 HEIDELBERGER, P., SHAHABUDDIN, P. and NICOLA, V. 1994. Bounded relative error in estimating transient measures of highly dependable non-Markovian sy stems. ACM Trans. Modeling Comp. Simulation 4 137 164.
  • 20 KESIDIS, G. and WALRAND, J. 1993. Quick simulation of ATM buffers with on-off multiclass Markov fluid sources. ACM Trans. Modeling Comp. Simulation 3 269 276.
  • 21 LEHTONEN, T. and Ny RHINEN, H. 1992. Simulating level-crossing probabilities by importance sampling. Adv. in Appl. Prob. 24 858 874.
  • 22 MCDONALD, D. R. 1995. Asy mptotics of first passage times for random walk in a quadrant. Working paper, Dept. Mathematics and Statistics, Univ. Ottawa.
  • 23 NAKAy AMA, M. K. 1994. A characterization of the simple failure biasing method for simulations of highly reliable Markovian sy stems. ACM Trans. Modeling Comp. Simulation 4 52 88.
  • 24 PAREKH, S. and WALRAND, J. 1989. A quick simulation method for excessive backlogs in networks of queues. IEEE Trans. Automat. Control 34 54 66.
  • 25 SADOWSKY, J. S. 1991. Large deviations and efficient simulation of excessive backlogs in a GI G m queue. IEEE Trans. Automat. Control 36 1383 1394.
  • 26 SADOWSKY, J. S. 1993. On the optimality and stability of exponential twisting in Monte Carlo estimation. IEEE Trans. Inform. Theory 39 119 128.
  • 27 SADOWSKY, J. S. 1997. On Monte Carlo estimation of large deviations probabilities. Unpublished manuscript.
  • 28 SADOWSKY, J. S. and BUCKLEW, J. A. 1990. On large deviations theory and asy mptotically efficient Monte Carlo estimation. IEEE Trans. Inform. Theory 36 579 588.
  • 29 SHAHABUDDIN, P. 1994. Importance sampling for the simulation of highly reliable Markovian sy stems. Management Science 40 333 352.
  • 30 SIEGMUND, D. 1976. Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673 684.
  • 31 SIEGMUND, D. 1985. Sequential Analy sis. Springer, New York.
  • NEW YORK, NEW YORK 10027 E-MAIL: pglasser@research.gsb.columbia.edu ywang@groucho.gsb.columbia.edu