Estimation non-paramétrique des quantiles extrêmes conditionnels

Alexandre Lekina 1
1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : The main goal of this thesis is to propose new estimators of extreme quantiles in the conditional case, that is to say in the situation where the variable of interest Y, supposed to be random and real, is recorded simultaneously with some covariate information X . To this aim, we focus on the case where the conditional distribution of Y given X = x is "heavy-tailed". Two situations are considered. First, when the covariate is deterministic and finite-dimensional or infinite-dimensional (i.e functional covariate), we propose to estimate the extreme quantiles by the "moving window approach". The asymptotic distribution of the proposed estimators is given in the case where the quantile is in the range of data or near and even beyond the sample. Next, when the covariate is random and finite-dimensional, we show that under some conditions, it is possible to estimate these extreme quantiles using a kernel estimator of the conditional survival function. As a consequence, this result allows us to introduce two smooth versions of the conditional tail index estimator necessary to extrapolate. Asymptotic distributions of these estimators are established. Furthermore, we also considered the case without covariate. When the underlying, the cumulative distribution function is "heavy-tailed". A new unconditional extreme quantile estimator is introduced and studied. To assess the behavior of all our new statistical tools, numerical experiments on simulated data are provided and illustrations on real datasets are presented.
Document type :
Mathématiques [math]. Université Joseph-Fourier - Grenoble I; Université de Grenoble, 2010. Français
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Submitted on : Thursday, February 26, 2015 - 7:36:27 PM
Last modification on : Wednesday, December 14, 2016 - 1:07:24 AM
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Alexandre Lekina. Estimation non-paramétrique des quantiles extrêmes conditionnels. Mathématiques [math]. Université Joseph-Fourier - Grenoble I; Université de Grenoble, 2010. Français. <tel-00529476v2>



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