Quelques contributions à la Théorie univariée des Valeurs Extrêmes et Estimation des mesures de risque actuariel pour des pertes à queues lourdes

El Hadji Deme 1, 2
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
2 Lerstad
LERSTAD - laboratoire d'Etudes et de recherches en Statistiques et Développement
Abstract : This thesis is divided into five chapters with an additional introduction and a conclusion. In the first chapter, we recall some basics on extreme value theory. In the second chapter, we consider a statistical processes depending on a continuous time tau and whose any margin can arise as a Generalized Hill's estimator.. This statistical process allows to discriminate completely the extremevalues domain of attraction. The asymptotic normality of this statistical processes was only given for tau > 1/2. We complete this study for 0 < tau < 1/2, and we give an approximation to the domain of attraction of Gumbel and Fréchet. Simulation studies carried out with the " R "software, can show the performance of these estimators. As an illustration, we propose an application of our methodology for hydraulic data. In the third chapter, we extend the study of the previous statistical processes to a functional framework. We propose a stochastic processes depending on a positive functional to obtain a large class of estimators of extreme values index which each estimator is a marginal one stochastic processes. The theoretical study of these stochastic process that we lead is based on the modern theory of functional weak convergence, which handle more complex estimatorsin the form of stochastic processes. We give the asymptotic distributions functions of these processes and we show that for some class, we have a non Gaussian asymptotic behavior, which will be fully characterized. In the fourth chapter, we are interested in the estimation of the second order parameter. Note that this parameter is of primordial importance in the adaptive choice of the best number of upper order statistics to be considered in the estimation of the extreme-value index. The estimation of the second order parameter can also be used to propose bias reduced estimators of the extreme value index, and has received a lot of attention in the extreme-value literature.. We propose a simple and general approach to estimate the second order parameter. It is shown that many estimators in the extreme-value literature can be read as particular cases of our approach. We illustrate how a lot of new asymptotically Gaussian estimators can be derived from this framework. Finally, some estimators are compared both from the asymptoticfinite sample size performances points of view. As an illustration, we propose a case study in the field of insurance. In the last chapter, we are interested in the actuarial risk premiums for some phenomena generating very significant financial losses (or extreme phenomenas). Many Risk measures or premium calcutation principles have been proposed in the actuarial literature. We focus on the adjusted risk premium.. Jones et Zitikis (2003) gave an estimate of the adjusted risk premium based on the empirical distribution and have established its asymptotic normality undersome appropriate conditions, and which are not fulfilled in the case of heavy-tailed distributions (case of extreme phenomena). we look at this framework and we consider a family of estimators of the ajusted risk premium based on the extreme values theory approach. We establish their asymptotic normality and we propose also an approach to bias reduction for these estimators. A simulation study is used to evaluate the quality of our estimators. As an illustration, we propose an application on real data insurance.
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El Hadji Deme. Quelques contributions à la Théorie univariée des Valeurs Extrêmes et Estimation des mesures de risque actuariel pour des pertes à queues lourdes. Méthodologie [stat.ME]. Université Gaston Berger, Saint Louis, Sénégal, 2013. Français. ⟨tel-01069382⟩



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