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

Annals of Applied Probability

Volumn 21, Issue 6, 2011, Pages 2315-2342

  Chan, Hock Peng ^a   Lai, Tze Leung ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Exceedance probabilities; Large deviations; Logarithmic efficiency; Sequential importance sampling and resampling

Indexed keywords

EID: 82655181334     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AAP758     Document Type: Article

References (22)

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. MR2434174
4. BROWN, L. (1986). Fundamentals of Statistical Exponential Families. Institute of Mathematical Statistics Lecture Notes 9. IMS, Hayward, CA.
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. MR1039183
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. MR1835035 (Pubitemid 33274477)
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. MR1970269
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. MR2308332
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. MR1890070 (Pubitemid 37023405)
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. MR1710982
12. DE LA PEÑA, V. H., LAI, T. L. and SHAO, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin. 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. MR2278333 (Pubitemid 43742903)
14. DEL MORAL, P. and GARNIER, J. (2005). Genealogical particle analysis of rare events. Ann. Appl. Probab. 15 2496-2534. MR2187302 (Pubitemid 43106162)
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. MR1847786
16. DUPUIS, P. and WANG, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1-38. MR2115034 (Pubitemid 40306622)
17. DUPUIS, P. and WANG, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Oper. Res. 32 723-757. MR2348245 (Pubitemid 351159480)
18. GLASSERMAN, P. andWANG, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7 731-746. MR1459268
19. NEY, P. andNUMMELIN, E. (1987).Markov Additive Processes. I. Eigenvalues Properties and Limit Theorems. II. Large Deviations. Ann. Probab. 15 561-592, 593-609. MR0885131
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. 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. MR1053850
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. MR0408141