Un modèle de Markov caché en assurance et Estimation de frontière et de point terminal

Abstract : This thesis is divided into two parts. We first focus on introducing a new model for loss processes in insurance: it is a process (J, N, S) where (J, N) is a Makov-modulated Poisson process and S is a process whose components are piecewise constant, nondecreasing processes. The increments of S are assumed to be conditionally independent given (J, N). Assuming further that the distribution of the jumps of S belongs to some parametric family, it is shown that the maximum likelihood estimator (MLE) of the parameters of this model is strongly consistent. An EM algorithm is given to help compute the MLE in practice. The method is used on real insurance data and its performances are examined on some finite sample situations. In an independent second part, the extreme-value theory problem of estimating the (finite) right endpoint of a cumulative distribution function F is considered: given a sample of independent copies of a random variable X with distribution function F, an estimator of the right endpoint of F is designed, using a high order moments method. We first consider the case when X is nonnegative, and the method is then generalised to the case when X is an arbitrary random variable with finite right endpoint by introducing another estimator. The asymptotic properties of both estimators are studied, and their performances are examined on some simulations. Based upon that work, an estimator of the frontier of the support of a random pair is constructed. Its asymptotic properties are discussed, and the method is compared to other classical methods in this framework.
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Gilles Stupfler. Un modèle de Markov caché en assurance et Estimation de frontière et de point terminal. Mathématiques [math]. Université de Strasbourg, 2011. Français. ⟨NNT : 2011STRA6117⟩. ⟨tel-00638368⟩

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