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Uniformly efficient simulation for extremes of Gaussian random fields

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  28 March 2018

Xiaoou Li  and
Gongjun Xu
Xiaoou Li*
Affiliation:
University of Minnesota
Gongjun Xu*
Affiliation:
University of Michigan
*
* Postal address: School of Statistics, University of Minnesota, 224 Church ST SE, Minneapolis, MN 55455, USA. Email address: [email protected]
** Postal address: Department of Statistics, University of Michigan, 1085 South University, Ann Arbor, MI 48109, USA. Email address: [email protected]
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Abstract

In this paper we consider the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problems. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.

Keywords

Rare eventimportance samplingGaussian random field

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60G15: Gaussian processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 157 - 178
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. Google Scholar
[2]Adler, R. J., Blanchet, J. and Liu, J. (2008). Efficient simulation for tail probabilities of Gaussian random fields. In Proceedings of the 2008 Winter Simulation Conference, IEEE, pp. 328–336. Google Scholar
[3]Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Prob. 22, 1167–1214. Google Scholar
[4]Adler, R. J., Müller, P. and Rozovskii, B. (1996). Stochastic Modelling in Physical Oceanography. Birkhaüser, Boston, MA. Google Scholar
[5]Adler, R. J., Taylor, J. E. and Worsley, K. J. (2017). Applications of random fields and geometry: foundations and case studies. Available at http://webee.technion.ac.il/Sites/People/adler/. Google Scholar
[6]Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York. Google Scholar
[7]Azaïs, J.-M. and Wschebor, M. (2008). A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stoch. Process. Appl. 118, 1190–1218. (Erratum: 120 (2010) 2100–2101.) Google Scholar
[8]Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ. Google Scholar
[9]Berman, S. M. (1985). An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Prob. 22, 454–460. Google Scholar
[10]Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216. Google Scholar
[11]Borell, C. (2003). The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337, 663–666. Google Scholar
[12]Cheng, D. and Xiao, Y. (2016). The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments. Ann. Appl. Prob. 26, 722–759. Google Scholar
[13]Dębicki, K., Hashorva, E. and Ji, L. (2016). Extremes of a class of nonhomogeneous Gaussian random fields. Ann. Prob. 44, 984–1012. Google Scholar
[14]Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Prob. 1, 66–103. CrossRefGoogle Scholar
[15]Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York. Google Scholar
[16]Glasserman, P. and Juneja, S. (2008). Uniformly efficient importance sampling for the tail distribution of sums of random variables. Math. Operat. Res. 33, 36–50. Google Scholar
[17]Juneja, S. and Shahabuddin, P. (2006). Rare-event simulation techniques: an introduction and recent advances. In Handbooks in Operations Research and Management Science, Vol. 13, Elsevier, pp. 291–350. Google Scholar
[18]Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhyā A 32, 369–378. Google Scholar
[19]Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin. CrossRefGoogle Scholar
[20]Li, X. and Liu, J. (2015). Rare-event simulation and efficient discretization for the supremum of Gaussian random fields. Adv. Appl. Prob. 47, 787–816. CrossRefGoogle Scholar
[21]Li, X., Liu, J. and Xu, G. (2016). On the tail probabilities of aggregated lognormal random fields with small noise. Math. Operat. Res. 41, 236–246. Google Scholar
[22]Li, X., Liu, J. and Ying, Z. (2016). Chernoff index for Cox test of separate parametric families. Preprint. Available at https://arxiv.org/abs/1606.08248. Google Scholar
[23]Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Prob. 40, 1069–1104. Google Scholar
[24]Liu, J. and Xu, G. (2012). Rare-event simulations for exponential integrals of smooth Gaussian processes. In Proceedings of the 2012 Winter Simulation Conference, IEEE, pp. 1–10. Google Scholar
[25]Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40, 262–293. Google Scholar
[26]Liu, J. and Xu, G. (2013). On the density functions of integrals of Gaussian random fields. Adv. Appl. Prob. 45, 398–424. CrossRefGoogle Scholar
[27]Liu, J. and Xu, G. (2014). Efficient simulations for the exponential integrals of Hölder continuous Gaussian random fields. ACM Trans. Model. Comput. Simul. 24, 9. Google Scholar
[28]Liu, J. and Xu, G. (2014). On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Prob. 24, 1691–1738. Google Scholar
[29]Marcus, M. B. and Shepp, L. A. (1970). Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151, 377–391. Google Scholar
[30]Sudakov, V. N. and Tsirelson, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. LOMI 41, 14–24. Google Scholar
[31]Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Prob. 21, 34–71. Google Scholar
[32]Talagrand, M. (1996). Majorizing measures: the generic chaining. Ann. Prob. 24, 1049–1103. Google Scholar
[33]Tsirelson, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan-USSR Symposium on Probability Theory, Springer, Berlin, pp. 20–41. Google Scholar
[34]Xu, G. (2014). Uniformly efficient simulation for tail probabilities of Gaussian random fields. In Proceedings of the 2014 Winter Simulation Conference, IEEE, pp. 533–542. Google Scholar