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Theses

Estimation d'une matrice de précision sous mélange de lois de Wishart

Abstract : In this thesis, we consider the problem of estimating the precision matrix \Sigma^{-1} of a mixture of Wishart distributions model S \mid V \sim \mathcal{W}_p(n,V \,\Sigma) under various Efron-Morris type losses, L_{k} \{ \Sigma^{-1},\hat\Sigma^{-1}\} = \tr [ \{\hat\Sigma^{-1} - \Sigma^{-1}\}^2 \, S^k ] , for k=1,2,3 \dots. Here S is the p \times p sample covariance matrix, V is a mixing variable which has a known distribution \mathcal{H}(\cdot) on \mathbb{R}_{+} and \mathcal{W}_p(n,V \,\Sigma) denotes the Wishart distribution with n degrees of freedom and positive-definite covariance matrix V \,\Sigma. In a unified approach to the cases where S is invertible and S is singular, we consider the class of usuel estimator a\,S^+ , where S^+ denotes the Moore-Penrose inverse of S (which coincides with the inverse S^{-1} of S when S is invertible) and a is a positive constant, we provide optimal estimators in this class, denoted by a^*\,S^+, under the losses L_1, L_2 and L_3 ; and alternative estimators, denoted by a_0\,S^+, under the losses L_0 and L_k, for k\ge 4, cases where optimal estimators do not exist. As the usual estimators of the form \hat\Sigma^{-1}_{a} = a\,S^+ are inadmissibles, we propose two types of alternative estimators of the form \hat\Sigma^{-1}_{a,c} = a\,S^{+} + c\,S\,G(S) and \hat\Sigma^{-1}_{a,c,r} = a\,S^{+} + c\,r(\tr \{S\}) \,S\,G(S), where c is a positive constant, r(\cdot) is a real function and G(S) is a p \times p homogeneous matrix function of order \alpha. Note that, for both types of estimators, the function G(S) may be orthogonally invariant or not. We develop an unbiased estimator of the risk difference between \hat \Sigma^{-1}_{a,c} (respectively \hat \Sigma^{-1}_{a,c,r}) and \hat \Sigma^{-1}_{a}, relying on a Stein-Haff type identity and we provide conditions on c, G(S) and r(\cdot) so that the estimators \hat\Sigma^{-1}_{a,c} and \hat\Sigma^{-1}_{a,c,r} improve on the estimators \hat\Sigma^{-1}_{a} under the losses L_{k} \{ \Sigma^{-1},\hat\Sigma^{-1}\} = \tr [ \{\hat\Sigma^{-1} - \Sigma^{-1}\}^2 \, S^k ] , for k=1,2,3 \dots. In particular, we give examples of Haff, Dey and Efron-Morris type estimators that dominate the estimators of the form \hat \Sigma^{-1}_a, specifically for a=a^* and a=a_0. Finally, based on the information provided by the estimators of Haff, Dey and Efron-Morris type whose correction function G(S) is homogeneous of degree -2, we highlight the fact that only this degree of homogeneity makes sense. We illustrate this phenomenon in the context of the estimation of a mean vector of a normal distribution by showing that, among the James-Stein type estimators of a certain degree of homogeneity, only the homogeneity of order -2 is valid.
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Submitted on : Wednesday, May 4, 2022 - 1:01:12 AM
Last modification on : Tuesday, May 31, 2022 - 4:49:10 PM

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Djamila Boukehil. Estimation d'une matrice de précision sous mélange de lois de Wishart. Statistiques [math.ST]. Normandie Université, 2021. Français. ⟨NNT : 2021NORMR053⟩. ⟨tel-03658435⟩

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