The Annals of Applied Probability

A sequential Monte Carlo approach to computing tail probabilities in stochastic models

Hock Peng Chan and Tze Leung Lai

Full-text: Open access

Enhanced PDF (307 KB)

Abstract

Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential Monte Carlo estimators, we show how resampling weights can be chosen to yield logarithmically efficient Monte Carlo estimates of large deviation probabilities for multidimensional Markov random walks.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 6 (2011), 2315-2342.

Dates
First available in Project Euclid: 23 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1322057323

Digital Object Identifier
doi:10.1214/10-AAP758

Mathematical Reviews number (MathSciNet)
MR2895417

Zentralblatt MATH identifier
1246.60042

Subjects
Primary: 60F10: Large deviations 65C05: Monte Carlo methods
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Exceedance probabilities large deviations logarithmic efficiency sequential importance sampling and resampling

Citation

Chan, Hock Peng; Lai, Tze Leung. A sequential Monte Carlo approach to computing tail probabilities in stochastic models. Ann. Appl. Probab. 21 (2011), no. 6, 2315--2342. doi:10.1214/10-AAP758. https://projecteuclid.org/euclid.aoap/1322057323


Export citation

References

  • [1] Baker, J. E. (1985). Adaptive selection methods for genetic algorithms. In Proc. International Conference on Genetic Algorithms and Their Applications (J. Grefenstette, ed.) 101–111. Erlbaum, Mahwah, NJ.
  • [2] Baker, J. E. (1987). Reducing bias and inefficiency in the selection algorithm. In Genetic Algorithms and Their Applications (J. Grefenstette, ed.) 14–21. Erlbaum, Mahwah, NJ.
  • [3] Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Probab. 18 1351–1378.
    Mathematical Reviews (MathSciNet): MR2434174
    Zentralblatt MATH: 1147.60315
    Digital Object Identifier: doi:10.1214/07-AAP485
    Project Euclid: euclid.aoap/1216677125
  • [4] Brown, L. (1986). Fundamentals of Statistical Exponential Families. Institute of Mathematical Statistics Lecture Notes 9. IMS, Hayward, CA.
    Mathematical Reviews (MathSciNet): MR882001
    Zentralblatt MATH: 0685.62002
  • [5] 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.
    Mathematical Reviews (MathSciNet): MR1039183
    Zentralblatt MATH: 0702.60028
    Digital Object Identifier: doi:10.2307/3214594
  • [6] Chan, H. P. and Lai, T. L. (2000). Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families. Ann. Statist. 28 1638–1669.
    Mathematical Reviews (MathSciNet): MR1835035
    Zentralblatt MATH: 1105.62367
    Digital Object Identifier: doi:10.1214/aos/1015957474
    Project Euclid: euclid.aos/1015957474
  • [7] Chan, H. P. and Lai, T. L. (2003). Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks. Ann. Appl. Probab. 13 395–429.
    Mathematical Reviews (MathSciNet): MR1970269
    Zentralblatt MATH: 1029.60058
    Digital Object Identifier: doi:10.1214/aoap/1050689586
    Project Euclid: euclid.aoap/1050689586
  • [8] Chan, H. P. and Lai, T. L. (2007). Efficient importance sampling for Monte Carlo evaluation of exceedance probabilities. Ann. Appl. Probab. 17 440–473.
    Mathematical Reviews (MathSciNet): MR2308332
    Zentralblatt MATH: 1134.65005
    Digital Object Identifier: doi:10.1214/105051606000000664
    Project Euclid: euclid.aoap/1174323253
  • [9] Chan, H. P. and Lai, T. L. (2008). A general theory of particle filters in hidden Markov models and some applications. Technical report, Dept. Statistics, Stanford Univ.
  • [10] Collamore, J. F. (2002). Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors. Ann. Appl. Probab. 12 382–421.
    Mathematical Reviews (MathSciNet): MR1890070
    Zentralblatt MATH: 1021.65003
    Digital Object Identifier: doi:10.1214/aoap/1015961169
    Project Euclid: euclid.aoap/1015961169
  • [11] Crisan, D., Del Moral, P. and Lyons, T. (1999). Discrete filtering using branching and interacting particle systems. Markov Process. Related Fields 5 293–318.
    Mathematical Reviews (MathSciNet): MR1710982
    Zentralblatt MATH: 0967.93088
  • [12] de la Peña, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2488094
  • [13] Del Moral, P., Doucet, A. and Jasra, A. (2006). Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 411–436.
    Mathematical Reviews (MathSciNet): MR2278333
    Zentralblatt MATH: 1105.62034
    Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00553.x
  • [14] Del Moral, P. and Garnier, J. (2005). Genealogical particle analysis of rare events. Ann. Appl. Probab. 15 2496–2534.
    Mathematical Reviews (MathSciNet): MR2187302
    Zentralblatt MATH: 1097.65013
    Digital Object Identifier: doi:10.1214/105051605000000566
    Project Euclid: euclid.aoap/1133965770
  • [15] Del Moral, P. and Jacod, J. (2001). Interacting particle filtering with discrete observations. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 43–75. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1847786
  • [16] Dupuis, P. and Wang, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1–38.
    Mathematical Reviews (MathSciNet): MR2115034
    Zentralblatt MATH: 1068.60036
    Digital Object Identifier: doi:10.1214/105051604000001016
    Project Euclid: euclid.aoap/1106922319
  • [17] Dupuis, P. and Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Oper. Res. 32 723–757.
    Mathematical Reviews (MathSciNet): MR2348245
    Zentralblatt MATH: 05279751
    Digital Object Identifier: doi:10.1287/moor.1070.0266
  • [18] Glasserman, P. and Wang, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7 731–746.
    Mathematical Reviews (MathSciNet): MR1459268
    Zentralblatt MATH: 0892.60043
    Digital Object Identifier: doi:10.1214/aoap/1034801251
    Project Euclid: euclid.aoap/1034801251
  • [19] Ney, P. and Nummelin, E. (1987). Markov Additive Processes. I. Eigenvalues Properties and Limit Theorems. II. Large Deviations. Ann. Probab. 15 561–592, 593–609.
    Mathematical Reviews (MathSciNet): MR885131
    Zentralblatt MATH: 0625.60027
    Digital Object Identifier: doi:10.1214/aop/1176992159
    Project Euclid: euclid.aop/1176992159
  • [20] Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1992). Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1201159
  • [21] Sadowsky, J. S. and Bucklew, J. A. (1990). On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Trans. Inform. Theory 36 579–588.
    Mathematical Reviews (MathSciNet): MR1053850
    Digital Object Identifier: doi:10.1109/18.54903
  • [22] Siegmund, D. (1975). Error probabilities and average sample number of the sequential probability ratio test. J. Roy. Statist. Soc. Ser. B 37 394–401.
    Mathematical Reviews (MathSciNet): MR408141

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more