Abstract
We present a method to obtain state- and time-dependent importance sampling estimators by repeatedly solving a minimum cross-entropy (MCE) program as the simulation progresses. This MCE-based approach lends a foundation to the natural notion to stop changing the measure when it is no longer needed. We use this method to obtain a state- and time-dependent estimator for the one-tailed probability of a light-tailed i.i.d. sum that is logarithmically efficient in general and strongly efficient when the jumps are Gaussian. We go on to construct an estimator for the two-tailed problem which is shown to be similarly efficient. We consider minor variants of the algorithm obtained via MCE, and present some numerical comparisons between our algorithms and others from the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Blanchet, J., & Glynn, P. (2006). Strongly efficient estimators for light-tailed sums. In L. Lenzini & R. Cruz (Eds.), Proceedings of the 1st international conference on performance evaluation methodologies and tools. New York: ACM.
Botev, Z. I., Kroese, D. P., & Taimre, T. (2007). Generalized cross-entropy methods with applications to rare-event simulation and optimization. Simulation: Transactions of the Society for Modeling and Simulation International, 83, 785–806.
Bucklew, J. A. (2004). Introduction to rare-event simulation. New York: Springer.
de Boer, P. T. (2006). Analysis of state-dependent importance sampling measures for the two-node tandem queue. ACM Transactions on Modeling Computer Simulation, 16, 225–250.
Dembo, A., & Zeitouni, O. (1998). Large deviations techniques and applications (2nd ed.). New York: Springer.
Dupuis, P., & Wang, H. (2004). Importance sampling, large deviations, and differential games. Stochastics and Stochastic Reports, 76, 481–508.
Dupuis, P., & Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Mathematics of Operations Research, 32, 723–757.
Ethier, S. N., & Kurtz, T. G. (1986). Markov processes: characterization and convergence. New York: Wiley.
Glassermann, P., & Wang, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Annals of Applied Probability, 7, 731–746.
Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. ACM Transactions on Modelling and Computer Simulation, 5, 43–85.
Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106, 620–630.
Jaynes, E. T. (1963). Information theory and statistical mechanics. In K. Ford (Ed.), Statistical physics (pp. 181–218). New York: Benjamin.
Kullback, S., & Khairat, M. A. (1966). A note on minimum discrimination information. Annals of Mathematical Statistics, 37, 279–280.
L’Ecuyer, P., Blanchet, J. H., Tuffin, B., & Glynn, P. W. (2008). Asymptotic robustness of estimators in rare-event simulation. ACM Transactions on Modeling and Computer Simulation (to appear).
Ridder, A., & Rubinstein, R. Y. (2007). Minimum cross-entropy methods for rare-event simulation. Simulation: Transactions of the Society for Modeling and Simulation International, 83, 769–784.
Rubinstein, R. Y. (2005). A stochastic minimum cross-entropy method for combinatorial optimization and rare-event estimation. Methodology and Computing in Applied Probability, 7, 5–50.
Rubinstein, R. Y., & Kroese, D. P. (2008). Simulation and the Monte Carlo method (2nd ed.). New York: Wiley.
Sadowsky, J. S., & Bucklew, J. A. (1990). On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Transactions on Information Theory, 36, 579–588.
Smith, P. J. (2001). Underestimation of rare event probabilities in importance sampling simulations. Simulation: Transactions of the Society for Modeling and Simulation International, 76, 140–150.
Tijms, H. C. (2003). A first course in stochastic models. New York: Wiley.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of T. Taimre supported by the Commonwealth Government of Australia, and by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ridder, A., Taimre, T. State-dependent importance sampling schemes via minimum cross-entropy. Ann Oper Res 189, 357–388 (2011). https://doi.org/10.1007/s10479-009-0611-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-009-0611-7