Extreme Value Analysis: an Introduction

Charras-Garrido, Myriam; Lezaud, Pascal

Journal de la société française de statistique, Tome 154 (2013) no. 2, pp. 66-97.

Nous donnons un aperçu des résultats probabilistes et statistiques utilisés dans la théorie des valeurs extrêmes, dont l’objectif est de prédire l’occurrence d’événements rares. Dans la première partie de l’article, nous expliquons que la distribution asymptotique des valeurs extrêmes appartient, dans un certain sens, à la famille des distributions des valeurs extrêmes généralisées qui dépendent d’un paramètre réel, appelé l’indice de valeur extrême. Dans la seconde partie, nous discutons des méthodes d’évaluation statistiques des queues basées sur l’estimation de l’indice des valeurs extrêmes.

We provide an overview of the probability and statistical tools underlying the extreme value theory, which aims to predict occurrence of rare events. Firstly, we explain that the asymptotic distribution of extreme values belongs, in some sense, to the family of the generalised extreme value distributions which depend on a real parameter, called the extreme value index. Secondly, we discuss statistical tail estimation methods based on estimators of the extreme value index.

[1] Adler, R. J. On excursion sets, tube formulas and maxima of random fields, Ann. Appl. Probab., Volume 10 (2000) no. 1, pp. 1-74 | Zbl

[2] Aldous, D. Probability Approximations via the Poisson Clumping Heuristic, Springer Verlag, 1989 | Zbl

[3] Azaïs, J. M.; Wschebor, M. Level sets and extrema of random processes and fields, John Wiley & Sons, 2009 | Zbl

[4] Beirlant, J.; Caeiro, F.; Gomes, M.I. An overview and open research topics in statistics of univariate extremes, REVSTAT - Statistical Journal, Volume 10 (2012) no. 1, pp. 1-31 | Zbl

[5] Beirlant, J.; Dierckx, G.; Goegebeur, Y.; Matthys, G. Tail Index Estimation and an Exponential Regression Model, Extremes, Volume 2 (1999) no. 2, pp. 177-200 | Zbl

[6] Beirlant, J.; Dierckx, G.; Guillou, A.; Staăricaă, C. On Exponential Representations of Log-Spacings of Extreme Order Statistics, Extremes, Volume 5 (2002) no. 2, pp. 157-180 | Zbl

[7] Berman, S. M. Sojourns and extremes of stochastic processes, Wadsworth and Brooks, 1992 | Zbl

[8] Beirlant, J.; Goegebeur, Y.; Segers, J.; Teugels, J. Statistics of Extremes, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd, 2004, 522 pages | Zbl

[9] Beirlant, J.; Goegebeur, Y.; Segers, J.; Teugels, J. Statistics of Extremes: Theory and Applications, Probability and Statistics, Wiley, 2004 | Zbl

[10] Bingham, N.H.; Goldie, C. M.; Teugels, J. L. Regular Variation, Cambridge University Press, 1989 | Zbl

[11] Billingsley, P. Probability and Measure, Wiley, 1995

[12] Bouleau, N. Splendeurs et misères des lois de valeurs extrêmes, Extremes (1991) ( http://halshs.archives-ouvertes.fr/docs/00/05/65/72/PDF/c15.pdf )

[13] Bucklew, J. Introduction to rare event simulation, Springer Series in Statistics, Springer-Verlag, 2011

[14] Castillo, E.; Hadi, A.S.; Balakrishnan, N.; Sarabia, J.M. Extreme value and related models with applications in engineering and science, Wiley, 2004, 362 pages | Zbl

[15] Coles, S. An introduction to statistical modeling of extreme values, Springer, 2001, 210 pages | Zbl

[16] Coles, S.; Powell, E.A. Bayesian Methods in Extreme Value Modelling: A Review and New Developments, International Statistical Review, Volume 64 (1996) no. 1, pp. 119-136 | Zbl

[17] Csorgo, S.; Viharos, L. Estimating the tail index (1998), pp. 833-881 | Zbl

[18] Davis, R.A. Limit laws for the maximum and minimum of stationary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Werwandtl Gebiete, Volume 61 (1982), pp. 31-42 | Zbl

[19] Doucet, A.; de Freitas, N.; Gordon, N. An Introduction to Sequential Monte Carlo Methods, Springer Verlag (2001) | Zbl

[20] Dekkers, A.L.M.; de Haan, L. On the Estimation of the Extreme-Value Index and Large Quantile Estimation, Annals of Statistics, Volume 17 (1989) no. 4, pp. 1795-1832 | Zbl

[21] Diebolt, J.; El-Aroui, M.A.; Garrido, M.; Girard, S. Quasi-conjugate Bayes estimates for GPD parameters and application to heavy tails modelling, Extremes, Volume 8 (2005) no. 1-2, pp. 57-78 | Zbl

[22] Dekkers, A.L.M.; Einmahl, J.H.J.; de Haan, L. A Moment Estimator for the Index of an Extreme-Value Distribution, Annals of Statistics, Volume 17 (1989) no. 4, pp. 1833-1855 | Zbl

[23] Diebolt, J.; Guillou, A.; Naveau, P.; Ribereau, P. Improving probability-weighted moment methods for the generalized extreme value distribution, REVSTAT - Statistical Journal, Volume 6 (2008) no. 1, pp. 33-50 | Zbl

[24] Diebolt, J.; Guillou, A.; Rached, I. Approximation of the distribution of excesses through a generalized probability-weighted moments method, Journal of Statistical Planning and Inference, Volume 137 (2007) no. 3, pp. 841-857 | Zbl

[25] de Haan, L.; Ferreira, A. Extreme value theory: An introduction, Springer, 2006, 418 pages | Zbl

[26] do Nascimento, F.F.; Gamerman, D.; Lopes, H.F. A semiparametric Bayesian approach to extreme value estimation, Statistics and Computing, Volume 22 (2011) no. 2, pp. 661-675 | Zbl

[27] Denzel, G.E.; O’Brien, G.L. Limit theorems for extreme values of chain-dependent processes, Annals of Probability, Volume 3 (1975), pp. 773-779 | Zbl

[28] Efron, B. Bootstrap methods: another look at the jackknife, The annals of Statistics, Volume 7 (1979) no. 7, pp. 1-26 | Zbl

[29] Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling Extremal Events for Insurance and Finance, Applications of Mathematics, 33, Springer, 2003

[30] Fraga Alves, M.I. A Location Invariant Hill-Type Estimator, Extremes, Volume 4 (2002) no. 3, pp. 199-217 | DOI | Zbl

[31] Feller, W. An Introduction to Probability Theory and Its Applications II, Wiley & Sons, 1971 | Zbl

[32] Feuerverger, A.; Hall, P. Estimating a tail exponent by modelling departure from a Pareto distribution, Annals of Statistics, Volume 27 (1999) no. 2, pp. 760-781 | Zbl

[33] Falk, M.; Hüsler, J.; Reiss, R. D. Laws of small numbers: extremes and rare events, Birkhäuser/Springer, 2011 | Zbl

[34] Fisher, R.A.; Tippett, L.H.C. Limiting forms of the frequency distribution of the largest and smallest member of a sample, Proc. Cambridge Phil. Soc., Volume 24 (1928), pp. 180-190 | JFM

[35] Guillou, A.; Hall, P. A diagnostic for selecting the threshold in extreme value analysis, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 63 (2001) no. 2, pp. 293-305 | DOI | Zbl

[36] Greenwood, J.A.; Landwehr, J.M.; Matalas, N.C.; Wallis, J.R. Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form, Water Ressources Research, Volume 15 (1979) no. 5, pp. 1049-1054

[37] Gomes, M.I.; Martins, M.J. “Asymptotically Unbiased” Estimators of the Tail Index Based on External Estimation of the Second Order Parameter, Extremes, Volume 5 (2002) no. 1, pp. 5-31 | Zbl

[38] Gomes, M.I.; Martins, M.J.; Neves, M. Improving second order reduced bias extreme value index estimation, REVSTAT - Statistical Journal, Volume 5 (2007) no. 2, pp. 177-207 | Zbl

[39] Gilleland, E.; Ribatet, M.; Stephenson, A.G. A software review for extreme value analysis, Extremes, Volume 16 (2013) no. 1, pp. 103-119 | DOI | Zbl

[40] Hill, B.M. A simple general approach to inference about the tail of a distribution, The annals of statistics, Volume 3 (1975) no. 5, pp. 1163-1174 | Zbl

[41] Hosking, J.R.M. Algorithm AS 215: Maximum-Likelihood Estimation of the Parameters of the Generalized Extreme-Value Distribution, Journal of the Royal Statistical Society: Series C (Applied Statistics), Volume 34 (2013) no. 3, pp. 301-310

[42] Hosking, J.R.M.; Wallis, J.R. Parameter and Quantile Estimation for the Generalized Pareto Distribution, Technometrics, Volume 29 (1987) no. 3, pp. 339-349 | DOI | Zbl

[43] Karamata, J. Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France, Volume 61 (1933), pp. 55-62 | JFM | Numdam | Zbl

[44] Korevaar, J. Tauberian Theory: a century of developments, A Series of Comprehensive Studies in Mathematics, 329, Springer, 2004 | Zbl

[45] Leadbetter, MR. On extreme values in stationary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Werwandtl Gebiete, Volume 28 (1974), pp. 289-303 | Zbl

[46] Leadbetter, MR. Extremes and local dependence in stationary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Werwandtl Gebiete, Volume 65 (1983), pp. 291-306 | Zbl

[47] Leadbetter, MR.; Lindgren, G.; Rootzén, H. Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, 1983 | Zbl

[48] Macleod, A.J. Algorithm AS 245: A robust and reliable algorithm for the logarithm of the gamma function, Journal of the Royal Statistical Society: Series C (Applied Statistics), Volume 38 (1989) no. 2, pp. 397-402 | Zbl

[49] Mason, D.M. Laws of large numbers for sums of extreme values, The Annals of Probability, Volume 10 (1982) no. 3, pp. 754-764 | Zbl

[50] O’Brien, G.L. The maximum term of uniformly mixing stationary processes, Zeitschrift für Wahrscheinlichkeitstheorie und Werwandtl Gebiete, Volume 30 (1974), pp. 57-63 | MR | Zbl

[51] O’Brien, G.L. Extreme values for stationary and Markov sequences, Annals of Probability, Volume 15 (1987), pp. 281-291 | MR | Zbl

[52] Pickands, J. Statistical inference using extreme order statistics, Annals of Statistics, Volume 3 (1975) no. 1, pp. 119-131 | MR | Zbl

[53] Prescott, P.; Walden, A.T. Maximum likelihood estimation of the parameters of the generalized extreme-value distribution, Biometrika, Volume 67 (1980) no. 3, pp. 723-724 | MR

[54] Prescott, P.; Walden, A.T. Maximum likelihood estimation of the parameters of the three-parameter generalized extreme-value distribution from censored samples, Journal of Statistical Computation and Simulation, Volume 16 (1983) no. 3-4, pp. 241-250 | DOI | Zbl

[55] Resnick, SI. Extreme Values, Regular Variation, and Point Processes, Springer Verlag, 1987 | MR

[56] Reiss, R.-D.; Thomas, M. Statistical analysis of extreme values: with applications to insurance, finance, hydrology and other fields, Birkhäuser, 2007, 511 pages | MR | Zbl

[57] Rare event simulation using Monte Carlo methods (Rubino, G.; Tuffin, B., eds.), John Wiley & Sons, 2009 | MR | Zbl

[58] Reiss, R. D.; Thomas, M. Statistical Analysis of Extreme Values, Birkhäuser, 1997 | MR | Zbl

[59] Segers, J. Generalized Pickands estimators for the extreme value index, Journal of Statistical Planning and Inference, Volume 128 (2005) no. 2, pp. 381-396 | DOI | MR | Zbl

[60] Smith, R.L. Maximum likelihood estimation in a class of nonregular cases, Biometrika, Volume 72 (1985) no. 1, pp. 67-90 | DOI | MR | Zbl

[61] Stephenson, A.; Tawn, J. Bayesian Inference for Extremes: Accounting for the Three Extremal Types, Extremes, Volume 7 (2004) no. 4, pp. 291-307 | MR | Zbl

[62] von Mises, R. La distribution de la plus grandes de $n$ valeurs, Reprinted(1954) in Selected Papers II: Amer. Math. Soc., Providence, R.I. (1975), pp. 271-294 | Zbl

[63] Zhou, C. Existence and consistency of the maximum likelihood estimator for the extreme value index, Journal of Multivariate Analysis, Volume 100 (2009) no. 4, pp. 794-815 | MR | Zbl

[64] Zhou, C. The extent of the maximum likelihood estimator for the extreme value index, Journal of Multivariate Analysis, Volume 101 (2010) no. 4, pp. 971-983 | MR | Zbl