no. 2
Numéro spécial : statistique des valeurs extrêmes
Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique
[Extreme quantile estimation for Weibull-tail distributions: an overview]
Journal de la société française de statistique, Volume 154 (2013) no. 2, pp. 98-118.

In this paper, an overview on extreme quantiles estimation for Weibull-tail distribution is provided. Recall that the survival function of a Weibull-tail distribution decreases exponentially fast. We show how this problem can be inserted in the more general setting of extreme value theory.

Cet article est une synthèse bibliographique des méthodes d’estimation de quantiles extrêmes pour les lois à queue de type Weibull. Ces lois ont une fonction de survie qui décroit vers zéro à une vitesse exponentielle. Nous montrons comment cette problématique s’inscrit plus largement dans la théorie des valeurs extrêmes.

Mot clés : Lois à queue de type Weibull, Synthèse bibliographique
Keywords: Weibull-tail distributions, Overview 
@article{JSFS_2013__154_2_98_0,
     author = {Gardes, Laurent and Girard, St\'ephane},
     title = {Estimation de quantiles extr\^emes pour les lois \`a queue de type {Weibull~:} une synth\`ese bibliographique},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {98--118},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {154},
     number = {2},
     year = {2013},
     mrnumber = {3120438},
     zbl = {1316.62064},
     language = {fr},
     url = {http://archive.numdam.org/item/JSFS_2013__154_2_98_0/}
}
TY  - JOUR
AU  - Gardes, Laurent
AU  - Girard, Stéphane
TI  - Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique
JO  - Journal de la société française de statistique
PY  - 2013
SP  - 98
EP  - 118
VL  - 154
IS  - 2
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2013__154_2_98_0/
LA  - fr
ID  - JSFS_2013__154_2_98_0
ER  - 
%0 Journal Article
%A Gardes, Laurent
%A Girard, Stéphane
%T Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique
%J Journal de la société française de statistique
%D 2013
%P 98-118
%V 154
%N 2
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2013__154_2_98_0/
%G fr
%F JSFS_2013__154_2_98_0
Gardes, Laurent; Girard, Stéphane. Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique. Journal de la société française de statistique, Volume 154 (2013) no. 2, pp. 98-118. http://archive.numdam.org/item/JSFS_2013__154_2_98_0/

[1] Asimit, V.; Li, D.; Peng, L. Pitfalls in using Weibull tailed distributions, Journal of Statistical Planning and Inference, Volume 140 (2010), pp. 2018-2024 | MR | Zbl

[2] Beirlant, J.; Broniatowski, M.; Teugels, J.L.; Vynckier, P. The mean residual life function at great age : applications to tail estimation, Journal of Statistical Planning and Inference, Volume 45 (1995), pp. 21-48 | MR | Zbl

[3] Beirlant, J.; Bouquiaux, C.; Werker, B. Semiparametric lower bounds for tail index estimation, Journal of Statistical Planning and Inference, Volume 136 (2006), pp. 705-729 | MR | Zbl

[4] Beirlant, J.; Dierckx, G.; Goegebeur, Y.; Matthys, G. Tail index estimation and an exponential regression model, Extremes, Volume 2 (1999), pp. 177-200 | MR | Zbl

[5] Beirlant, J.; Dierckx, G.; Guillou, A.; Stǎricǎ, C. On exponential representations of log-spacings of extreme order statistics, Extremes, Volume 5 (2002), pp. 157-180 | MR | Zbl

[6] Berred, M. Record values and the estimation of the Weibull tail-coefficient, Comptes-Rendus de l’Académie des Sciences, Volume T. 312, Série I (1991), pp. 943-946 | MR | Zbl

[7] Bingham, N.H.; Goldie, C.M.; Teugels, J.L. Regular Variation, Cambridge University Press, 1987 | MR | Zbl

[8] Brodin, E.; Rootzén, H. Univariate and Bivariate GPD Methods for Predicting Extreme Wind Storm Losses, Insurance : Mathematics and Economics, Volume 44 (2009), pp. 345-356 | MR | Zbl

[9] Broniatowski, M. On the estimation of the Weibull tail coefficient, Journal of Statistical Planning and Inference, Volume 35 (1993), pp. 349-366 | MR | Zbl

[10] Beirlant, J.; Teugels, J.L. Modelling large claims in non-life insurance, Insurance : Mathematics and Economics, Volume 11 (1992), pp. 17-29 | MR | Zbl

[11] Beirlant, J.; Teugels, J.L.; Vynckier, P. Practical analysis of extreme values, Leuven University Press, Leuven, Belgium, 1996 | Zbl

[12] Csörgő, S.; Deheuvels, P.; Mason, D.M. Kernel estimates of the tail index of a distribution, The Annals of Statistics, Volume 13 (1985), pp. 1050-1077 | MR | Zbl

[13] Csörgő, S.; Viharos, L. Estimating the tail index, Asymptotic Methods in Probability and Statistics (Szyszkowicz, B., ed.), TEST, North-Holland, Amsterdam, 1998, pp. 833-881 | MR | Zbl

[14] Dierckx, G.; Beirlant, J.; Waal, D. De; Guillou, A. A new estimation method for Weibull-type tails based on the mean excess function, Journal of Statistical Planning and Inference, Volume 139(6) (2009), pp. 1905-1920 | MR | Zbl

[15] Diebolt, J.; Gardes, L.; Girard, S.; Guillou, A. Bias-reduced estimators of the Weibull tail-coefficient, Test, Volume 17 (2008), pp. 311-331 | MR | Zbl

[16] Diebolt, J.; Gardes, L.; Girard, S.; Guillou, A. Bias-reduced extreme quantiles estimators of Weibull tail-distributions, Journal of Statistical Planning and Inference, Volume 138 (2008), pp. 1389-1401 | MR | Zbl

[17] de Haan, L.; Ferreira, A. Extreme value theory : an introduction, Springer, 2006 | MR | Zbl

[18] Ditlevsen, O. Distribution arbitrariness in structural reliability, Structural Safety and Reliability (Balkema, ed.), TEST, Rotterdam, 1998, pp. 1241-1247

[19] Drees, H.; Kaufmann, E. Selecting the optimal sample fraction in univariate extreme value estimation, Stochastic process and Application, Volume 75 (1998), pp. 149-172 | MR | Zbl

[20] Embrechts, P.; Klüppelberg, C.; Mikosch, T. Modelling extremal events, Springer, 1997 | MR

[21] Falk, M. Some best parameter estimates for distributions with finite endpoint, Statistics, Volume 27 (1995), pp. 115-125 | MR | Zbl

[22] Feuerverger, A.; Hall, P. Estimating a Tail Exponent by Modelling Departure from a Pareto Distribution, The Annals of Statistics, Volume 27 (1999), pp. 760-781 | MR | Zbl

[23] Falk, M.; Marohn, F. Efficient estimation of the shape parameter in Pareto models with partially known scale, Statistics & Decisions, Volume 15 (1997), pp. 229-239 | MR | Zbl

[24] Goegebeur, Y.; Beirlant, J.; de Wet, T. Generalized kernel estimators for the Weibull-tail coefficient, Communications in Statistics - Theory and Methods, Volume 39 (2010) no. 20, pp. 3695-3716 | MR | Zbl

[25] Gardes, L.; Girard, S. Estimating extreme quantiles of Weibull tail-distributions, Communication in Statistics - Theory and Methods, Volume 34 (2005), pp. 1065-1080 | MR | Zbl

[26] Gardes, L.; Girard, S. Comparison of Weibull tail-coefficient estimators, REVSTAT - Statistical Journal, Volume 4(2) (2006), pp. 163-188 | MR | Zbl

[27] Gardes, L.; Girard, S. Estimation of the Weibull tail-coefficient with linear combination of upper order statistics, Journal of Statistical Planning and Inference, Volume 138 (2008), pp. 1416-1427 | MR | Zbl

[28] Goegebeur, Y.; Guillou, A. Goodness-of-fit testing for Weibull-type behavior, Journal of Statistical Planning and Inference, Volume 140(6) (2010), pp. 1417-1436 | MR | Zbl

[29] Gardes, L.; Girard, S.; Guillou, A. Weibull tail-distributions revisited : a new look at some tail estimators, Journal of Statistical Planning and Inference, Volume 141 (2011) no. 1, pp. 429-444 | MR | Zbl

[30] Girard, S.; Guillou, A.; Stupfler, G. Estimating an endpoint with high order moments in the Weibull domain of attraction, Statistics and Probability Letters, Volume 82 (2012), pp. 2136-2144 | MR

[31] Geluk, J.L.; Haan, L. De Regular variation, extensions and Tauberian theorems, Center for Mathematics and Computer Science, Amsterdam, Netherlands, 1987 | MR | Zbl

[32] Girard, S. A Hill type estimate of the Weibull tail-coefficient, Communication in Statistics - Theory and Methods, Volume 33(2) (2004), pp. 205-234 | MR | Zbl

[33] Gomes, M.I.; Martins, M.J.; Neves, M. Improving second order reduced bias extreme value index estimation, REVSTAT - Statistical Journal, Volume 5(2) (2007), pp. 177-207 | MR | Zbl

[34] Gnedenko, B. Sur la distribution limite du terme maximum d’une série aléatoire, The Annals of Mathematics, Volume 44 (1943), pp. 423-453 | MR | Zbl

[35] Gomes, M.I. Asymptotic unbiased estimators of the tail index based on external estimation of the second order parameter, Extremes, Volume 5(1) (2002), pp. 5-31 | MR | Zbl

[36] Hill, B.M. A simple general approach to inference about the tail of a distribution, The Annals of Statistics, Volume 3 (1975), pp. 1163-1174 | MR | Zbl

[37] Hall, P.; Park, B.U. New methods for bias correction at endpoints and boundaries, The Annals of Statistics, Volume 30(5) (2002), pp. 1460-1479 | MR | Zbl

[38] Kratz, M.; Resnick, S. The QQ-estimator and heavy tails, Stochastic Models, Volume 12 (1996), pp. 699-724 | MR | Zbl

[39] Lepski, O.V.; Mammen, E.; Spokoiny, V.G. Optimal spatial adaption to inhomogeneous smoothness : an approach based on kernel estimates with variable bandwidth selectors, The Annals of Statistics, Volume 25 (1997), pp. 929-947 | MR | Zbl

[40] Mason, D.M. Asymptotic normality of linear combinations of order statistics with a smooth score function, The Annals of Statistics, Volume 9(4) (1981), pp. 899-908 | MR | Zbl

[41] Methni, J. El; Gardes, L.; Girard, S.; Guillou, A. Estimation of extreme quantiles from heavy and light tailed distributions, Journal of Statistical Planning and Inference, Volume 142 (2012) no. 10, pp. 2735-2747 | MR | Zbl

[42] Mercadier, C.; Soulier, P. Optimal rates of convergence in the Weibull model based on kernel-type estimators, Statistics and Probability Letters, Volume 82 (2011), pp. 548-556 | MR | Zbl

[43] Peng, L.; Yongcheng, Q. Estimating the first and second order parameters of a heavy tailed distribution, Australian and New Zealand Journal of Statistics, Volume 46(2) (2004), pp. 305-312 | MR | Zbl

[44] Resnick, S.I. Extreme values, regular variation and point processes, Springer Series in Operations Research and Financial Engineering, 1987 | MR | Zbl

[45] Rootzén, H.; Tajvidi, T. Can losses caused by wind storms be predicted from meteorological observations ?, Scandinavian Actuarial Journal, Volume 5 (2001), pp. 162-175 | MR | Zbl

[46] Schultze, J.; Steinebach, J. On least squares estimates of an exponential tail coefficient, Statistics and Decisions, Volume 14 (1996), pp. 353-372 | MR | Zbl

[47] Weissman, I. Estimation of parameters and large quantiles based on the k largest observations, Journal of the American Statistical Association, Volume 73 (1978), pp. 812-815 | MR | Zbl