Estimation des limites d'extrapolation par les lois de valeurs extrêmes. Application à des données environnementales.

Clément Albert 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 : This thesis takes place in the extreme value statistics framework. It provides three main contributions to this area. Extreme quantile estimation is a two step approach. First, it consists in proposing an extreme value based quantile approximation. Then, estimators of the unknown quantities are plugged in the previous approximation leading to an extreme quantile estimator. The first contribution of this thesis is the study of the extrapolation error, which is the error due to the extreme value based approximation of the true quantile. These investigations are carried out using two different kind of estimators, both based on the well-known Generalized Pareto approximation : the Exponential Tail estimator dedicated to the Gumbel maximum domain of attraction and theWeissman estimator dedicated to the Fréchet one. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In constrast, the extrapolation error associated with Weissman estimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are then derived and their accuracy is illustrated numerically. The second contribution is the proposition of a new extreme quantile estimator. The problem is addressed in the framework of the so-called "log-Generalized Weibull tail limit" model, where the logarithm of the inverse cumulative hazard rate function is supposed to be of extended regular variation. Based on this model, estimators of the parameters are proposed. Then, a new estimator of extreme quantiles is derived from the latter. Its asymptotic normality is established and its behavior in practice is illustrated on both real and simulated data. The third contribution of this thesis is the proposition of new mathematical tools allowing the quantification of extrapolation limits associated with a real dataset. These tools consist in some estimators of the extrapolation error. To build them, we take advantages on one hand of the first study we did by proposing first order approximations which are widely applicable in practice. On the other hand, we use the proposed estimators of the "log-Generalized Weibull tail limit" model to estimate the previous approximations. Performances of the obtained estimators are illustrated on simulated data. These estimators are finally used to estimate the extrapolation limits associated with three real datasets consisting in daily measures of some environmental variables. Depending on the climatic phenomena, we show that the extrapolation limits can be more or less stringent.
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Submitted on : Friday, February 1, 2019 - 2:24:37 PM
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Clément Albert. Estimation des limites d'extrapolation par les lois de valeurs extrêmes. Application à des données environnementales.. Mathématiques [math]. Communauté Universite Grenoble Alpes, 2018. Français. ⟨tel-01971408v2⟩

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