Peaks-Over-Threshold Modeling Under Random Censoring |
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Authors: | Jan Beirlant Gwladys Toulemonde |
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Institution: | 1. Department of Mathematics and Leuven Statistics Center , University of Leuven , France;2. I3M, Université de Montpellier 2 , Montpellier, France |
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Abstract: | Recently, the topic of extreme value under random censoring has been considered. Different estimators for the index have been proposed (see Beirlant et al., 2007
Beirlant , J. ,
Guillou , A. ,
Dierckx , G. ,
Fils-Villetard , A. ( 2007 ). Estimation of the extreme value index and extreme quantiles under random censoring . Extremes 10 : 151 – 174 .Crossref] , Google Scholar]). All of them are constructed as the classical estimators (without censoring) divided by the proportion of non censored observations above a certain threshold. Their asymptotic normality was established by Einmahl et al. (2008
Einmahl , J. H. J. ,
Fils-Villetard , A. ,
Guillou , A. ( 2008 ). Statistics of extremes under random censoring . Bernoulli 14 ( 1 ): 207 – 227 . Google Scholar]). An alternative approach consists of using the Peaks-Over-Threshold method (Balkema and de Haan, 1974
Balkema , A. ,
de Haan , L. ( 1974 ). Residual life at great age . Ann. Probab. 2 : 792 – 804 .Crossref], Web of Science ®] , Google Scholar]; Smith, 1987
Smith , R. L. ( 1987 ). Estimating tails of probability distributions . Ann. Statist. 15 : 1174 – 1207 .Crossref], Web of Science ®] , Google Scholar]) and to adapt the likelihood to the context of censoring. This leads to ML-estimators whose asymptotic properties are still unknown. The aim of this article is to propose one-step approximations, based on the Newton-Raphson algorithm. Based on a small simulation study, the one-step estimators are shown to be close approximations to the ML-estimators. Also, the asymptotic normality of the one-step estimators has been established, whereas in case of the ML-estimators it is still an open problem. The proof of our result, whose approach is new in the Peaks-Over-Threshold context, is in the spirit of Lehmann's theory (1991
Lehmann , E. L. ( 1991 ). Theory of Point Estimation . Pacific Grove , CA : Wadsworth & Brooks/Cole Advanced Books & Software .Crossref] , Google Scholar]). |
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Keywords: | Asymptotic normality Extreme value index Newton–Rapshon algorithm Random censoring |
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