Comptes Rendus
no. 3
Statistics/Probability Theory
Asymptotic normality of the ET method – application to the ET test
[Normalité asymptotique de la méthode ET – application au test ET]

Présenté par : Paul Deheuvels

Jean Diebolt1 ; Myriam Garrido2 ; Stéphane Girard2
1 CNRS, Université de Marne-la-Vallée, 5, bd Descartes, 77454 Marne-la-Vallée cedex 2, France
2 INRIA Rhône-Alpes, projet IS2, ZIRST, 655, avenue de l'Europe, Montbonnot, 38334 Saint-Ismier cedex, France
Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 213-218.

Nous proposons un test permettant de vérifier l'adéquation de la queue d'une fonction de répartition F0 aux observations extrêmes et de contrôler si cette queue fournit des extrapolations raisonnables au-delà de l'observation maximale. Le test est basé sur la loi asymptotique de l'estimateur ET (Exponential Tail) des quantiles extrêmes qui est établie dans une Note jointe. Le niveau et la puissance asymptotiques du test sont étudiés pour plusieurs classes de lois.

We propose a procedure to test the adequacy of the tail of a given F0 to extreme observations and to check that this tail provides reasonable extrapolations above the maximal observation. The test is based on the asymptotic distribution of the ET (Exponential Tail) estimate of extreme quantiles which is established in a companion Note. The asymptotic level and power of the test are studied for several classes of distributions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00238-3
Jean Diebolt 1 ; Myriam Garrido 2 ; Stéphane Girard 2

1 CNRS, Université de Marne-la-Vallée, 5, bd Descartes, 77454 Marne-la-Vallée cedex 2, France
2 INRIA Rhône-Alpes, projet IS2, ZIRST, 655, avenue de l'Europe, Montbonnot, 38334 Saint-Ismier cedex, France 
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     author = {Jean Diebolt and Myriam Garrido and St\'ephane Girard},
     title = {Asymptotic normality of the {ET} method {\textendash} application to the {ET~test}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {213--218},
     publisher = {Elsevier},
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Jean Diebolt; Myriam Garrido; Stéphane Girard. Asymptotic normality of the ET method – application to the ET test. Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 213-218. doi : 10.1016/S1631-073X(03)00238-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00238-3/

[1] N.H. Bingham; C.M. Goldie; J.L. Teugels Regular Variation, Encyclopedia Math. Appl., 27, Cambridge University Press, 1987

[2] R. Davis; S. Resnick Tail estimates motivated by extreme value theory, Ann. Statist., Volume 12 (1984), pp. 1467-1487

[3] J. Diebolt; S. Girard Consistency of the ET method and smooth variations, C. R. Acad. Sci. Paris, Sér. I, Volume 329 (1999), pp. 821-826

[4] J. Diebolt, M. Garrido, S. Girard, Le test ET : test d'adéquation d'un modèle central à une queue de distribution, Technical Report INRIA RR-4170, 2001

[5] J. Diebolt, M. Garrido, S. Girard, Asymptotic normality of the ET method for extreme quantile estimation. Application to the ET test, Technical Report INRIA RR-4551, 2002

[6] J. Diebolt, M. Garrido, S. Girard, Asymptotic normality of the ET method for extreme quantile estimation, C. R. Acad. Sci. Paris, Sér. I (2002) submitted for publication

[7] O. Ditlevsen Distribution arbitrariness in structural reliability, Structural Safety and Reliability, Balkema, Rotterdam, 1994, pp. 1241-1247

[8] G. Hahn; W. Meeker Pitfalls and practical considerations in product life analysis, part 1: Basic concepts and dangers of extrapolation, J. Quality Technology, Volume 14 (1982)

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