no. 4
Estimation de probabilités et de quantiles rares pour la caractérisation d’une zone de retombée d’un engin
Journal de la société française de statistique, Tome 152 (2011) no. 4, pp. 1-29.

Afin de quantifier les risques ainsi qu’évaluer des performances de système, il est souvent nécessaire d’estimer des quantiles et des probabilités faibles. Les techniques habituelles d’estimation de type méthode de Monte Carlo n’étant plus efficaces, nous détaillons les principales techniques d’estimation de probabilités rares telles que l’importance sampling, l’importance splitting ou la théorie des valeurs extrêmes. Ces différents algorithmes sont appliqués au cas de l’estimation d’une zone de retombée d’un engin spatial.

Estimating rare event probability and quantile with a valuable accuracy is an important source of interest in reliability and safety. Since usual estimation techniques such as Monte Carlo method are not efficient for low probabilities, different methods have been investigated: importance splitting, importance sampling or extreme value theory. These algorithms are compared and then applied to the safety zone estimation of an aerospace vehicle.

Mot clés : Estimation d’évènements rares, Théorie des valeurs extrêmes, Importance sampling, Importance splitting
Keywords: Rare event estimation, Extreme value theory, Importance sampling, Importance splitting 
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Morio, Jérôme; Pastel, Rudy; Le Gland, François. Estimation de probabilités et de quantiles rares pour la caractérisation d’une zone de retombée d’un engin. Journal de la société française de statistique, Tome 152 (2011) no. 4, pp. 1-29. http://www.numdam.org/item/JSFS_2011__152_4_1_0/

[1] Botev, Z. I; Kroese, D.P.; Taimre, T. Generalized Cross-Entropy Methods with Applications to Rare-Event simulation and Optimization, Simulation, Volume 11 (2007), pp. 785 -806

[2] Borcherds, P. H. Importance Sampling : An illustrative introduction, European Journal of Physics (2000), pp. 405-411

[3] Beichl, I.; Sullivan, F. The importance of Importance Sampling, Computing in science and engineering, Volume 1 (1999), pp. pp.71-73

[4] Bucklew, J. A. Large Deviation Techniques in Decision, Simulation, and Estimation, John Wiley, New York (1990)

[5] Cerou, F.; Del Moral, P.; Furon, T.; Guyader, A. Sequential Monte Carlo for rare event estimation, Statistics and Computing (2011) | Zbl

[6] Cerou, F.; Guyader, A. Adaptive particle techniques and rare event estimation, Volume 19, ESAIM  : Proceedings (2007), pp. 65-72 | Zbl

[7] Cannamela, C.; Garnier, J.; Iooss, B. Controlled stratification for quantile estimation, Annals of Applied Stats, Volume 2 (2008), pp. 1554-1580 | Zbl

[8] Coles, S.G. An introduction to statistical modeling of extreme values, Springer, New York (2001) | Zbl

[9] Daniels, H. E. Saddlepoint approximations in statistic, Ann. Math. Statis, Volume 5 (1954), pp. 631-650 | Zbl

[10] Denny, M. Introduction to importance Sampling in rare-event simulations, European Journal of Physics (2001), pp. 403-411

[11] de Haan, L.; Pickands, J. Stationary min-stable stochastic processes, Probability Theory and Related Fields, Volume 7 (1986), pp. 477-492 | Zbl

[12] Davison, A.C.; Smith, R.L. Models for exceedances over high thresholds (with discussion), Journal of the Royal Statistical Society, Volume 52 (1990), pp. 393-442 | Zbl

[13] Embrechts, P.; Kluppelberg, C.; Mikosch, T. Modelling Extremal Events for Insurance and Finance, Springer Verlag, Berlin (1997) | Zbl

[14] Fisher, R.A.; Tippett, L.H.C. On the estimation of the frequency distributions of the largest or smallest member of a sample, Proceedings of the Cambridge Philosophical Society (1928) | JFM

[15] Glasserman, P.; Heidelberger, P.; Shahabuddin, P.; Zajic, T. Splitting for rare event simulation : analysis of simple cases, Proceeding of the 1996 Winter Simulation Conference (1996), pp. 302-308

[16] Glad, I. K.; Hjort, N. L.; Ushakov, N. G. Mean-squared error of kernel estimators for finite values of the sample size, Journal of Mathematical Sciences, Volume 146 (2007), pp. 5977-5983

[17] Gilli, M.; Këllezi, E. An Application of Extreme Value Theory for Measuring Risk, Computationnal Economics, Volume 27 (2006), pp. 207-228 | Zbl

[18] Glynn, P. Importance sampling for Monte Carlo estimation of quantiles, Proceedings of the Second International Workshop on Mathematical Methods in Stochastic Simulation and Experimental Design, Saint Petersburg, Publishing House of Saint Petersburg University (1996), pp. 180-185

[19] Hesterberg, T.C.; Nelson, B.L. Control variates for probability and quantile estimation, Management Science, Volume 44 (1998) no. 9, pp. 1295-1312 | Zbl

[20] Homem-de-Mello, T.; Rubinstein, R. Y. Estimation of rare event probabilities using cross–entropy, Proceedings of the 2002 Winter Simulation Conference, San Diego (2002), pp. 310-319

[21] Hosking, J.R.M.; Wallis, J.R. Parameter and quantile estimation for the Generalized Pareto Distribution, Technometrics, Volume 29 (1987), pp. 339-349 | Zbl

[22] Lagnoux, A. Rare event simulation, Probability in the Engineering and Informational science, Volume 20 (2006), pp. 45-66 | Zbl

[23] L’Ecuyer, P.; Demers, V.; Tuffin, B. Splitting for rare event simulation, Proceeding of the 2006 Winter Simulation Conference, Monterey (2006), pp. 137-148

[24] L’Écuyer, P.; Le Gland, F.; Lezaud, P.; Tuffin, B. Splitting methods, Monte Carlo Methods for Rare Event Analysis (Rubino, Gerardo; Tuffin, Bruno, eds.), John Wiley & Sons, Chichester, 2009, pp. 39-61 | Zbl

[25] Leadbetter, M.R.; Lindgren, G.; Rootzén, H. Extremes and related properties of random sequences and series, Springer Verlag, New York, USA (1983) | Zbl

[26] Lugannani, R.; Rice, S. Saddlepoint approximation for the distribution of the sum of independent random variables, Adv. in Appl. Probab, Volume 12 (1980), pp. 475-490 | Zbl

[27] Leadbetter, M.R.; Rootzén, H. Extremal theory for stochastic processes, Annals of Probability, Volume 16 (1988), pp. 431-478 | Zbl

[28] Mikhailov, G. A. Parametric Estimates by the Monte Carlo Method, VSP, Utrecht (NED) (1999) | Zbl

[29] Millet, A. Méthode de Monte Carlo, Université Paris VI et Paris VII, Polycopié (2003)

[30] Molchanov, I. S. Empirical estimation of distribution quantiles of random closed sets, Theory Proba. Appli., Volume 35 (1990), pp. 594-600 | Zbl

[31] Morio, J. How to approach the importance sampling density, Eur. J. Phys, Volume 31 (2010), pp. 41-48 | Zbl

[32] Morio, J. Non parametric adaptive importance sampling of the probability estimation of launcher impact position, Reliability Engineering and System Safety, Volume 96 (2011) no. 1, pp. 178-183

[33] Morio, J.; Pastel, R. Sampling technique for launcher impact safety zone estimation, Acta Astronautica, Volume 66 (2010) no. 5-6, pp. 736-741

[34] Morio, J.; Pastel, R.; Le Gland, F. An overview of importance splitting for rare event simulation, European Journal of Physics, Volume 31 (2010), pp. 1295-1303

[35] McNeil, A.; Saladin, T. The peaks over threshold method for estimating high quantiles of loss distributions, Proceedings of the 28th International ASTIN Colloquium,Cairns (1997)

[36] Niederreiter, H.; Spanier, J. Monte Carlo and Quasi-Monte Carlo Methods, Springer (2000) | Zbl

[37] Oakley, J. Estimating percentiles of uncertain computer code outputs, Appl. Statist., Volume 53 (2004), pp. 83-93 | Zbl

[38] Picheny, V.; Ginsbourger, D.; Roustant, O.; Haftka, R.T. Adaptive designs of experiments for accurate approximation of a target region, J. Mech. Des., Volume 137 (2010) no. 7

[39] Pickands, J. Statistical inference using extreme order statistics, Annals of Statistics, Volume 3 (1975), pp. 119-131 | Zbl

[40] Piera Martinez, M.; Vazquez, E.; Walter, E.; Fleury, G.; Kielbasa, R. Estimation of extreme values, with application to uncertain systems, Proceedings of the 14th IFAC Symposium on System Identification (SYSID), Newcastle, IFAC (2006), pp. 1027-1032

[41] Ranjan, P.; Bingham, D.; Michailidis, G. Sequential experiment design for contour estimation from complex computer codes, Technometrics, Volume 50 (2008), pp. 527-541

[42] Robert, C. P.; Casella, G. Monte Carlo Statistical Methods, Springer, New York (2005) | Zbl

[43] Rubinstein, R.Y.; Kroese, D.P. The Cross-Entropy Method : A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning, Springer Verlag, New York (2004) | Zbl

[44] Rubinstein, R.Y. Simulation and the Monte Carlo Method, John Wiley (1981) | Zbl

[45] Rubinstein, R.Y. Optimization of computer simulation models with rare events, European Journal of Operations Research, Volume 99 (1997), pp. 89-112 | Zbl

[46] Rubinstein, R.Y. The cross-entropy method for combinatorial and continuous optimization, Methodology and Computing in Applied Probability, Volume 2 (1999), pp. 127-129 | Zbl

[47] Rutherford, B. A response-modeling alternative to surrogate models for support in computational analyses, Reliability Engineering and System Safety, Volume 91 (2006), pp. 1322-1330

[48] Silverman, B.W. Density Estimation for Statistics and Data Analysis, London : Chapman and Hall (1986) | Zbl

[49] Sobol, I. M. A Primer for the Monte Carlo Method, CRC Press, Boca Raton, Fl. (1994) | Zbl

[50] Tierney, L. Markov chains for exploring posterior distributions, Annals of Statistics, Volume 22 (1994), pp. 1701-1762 | Zbl

[51] Vazquez, E.; Bect, J. A sequential Bayesian algorithm to estimate a probability of failure, Proceedings of the 15th IFAC Symposium on System Identification (SYSID), Saint–Malo, IFAC (2009), pp. 546-550

[52] Zhang, P. Nonparametric importance sampling, Journal of the American Statistical Association, Volume 91 (1996) no. 434, pp. 1245-1253 | Zbl