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Information auxiliaire non paramétrique : géometrie de l'information, processus empirique et applications

Abstract : The first aim of this thesis was to develop a method for the optimal use of auxiliary information, i.e. information external to the observed statistical experiment. Thus we propose to search for the discrete probability measure that has as support the sample verifying the auxiliary information and that is the closest to the empirical measure in the sense of the information geometry. We prove that there are two solutions to this problem and we show that there is also a common approximation of these two solutions allowing to define the informed empirical measure. Subsequently, we study the properties of this informed empirical measure by establishing non-asymptotic concentration and asymptotic results of the P-Glivenko-Cantelli and P-Donsker type. The major consequence of accurate information is the uniform decrease in the asymptotic variance of the empirical process. The second aim of this thesis was to generalize the empirical informed measure to so-called weak auxiliary information (e.g. expert preference). After generalizing the empirical informed measure to these weak auxiliary information, we establish asymptotic results of the type P-Glivenko-Cantelli and P-Donsker. The third aim of this thesis was to study the impact of using false auxiliary information and to set up an adaptive procedure to select the relevant auxiliary information for the estimation of a parameter of interest. Finally, we conclude this thesis by proposing applications of the informed empirical measure to various problems in statistics and stochastic simulation (use of auxiliary variables, improvement of the Monte Carlo method, informed maximum likelihood, least squares estimator with auxiliary information, learning, etc). These applications illustrate the positive and relatively immediate impact of auxiliary information.
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Submitted on : Thursday, September 22, 2022 - 11:52:24 AM
Last modification on : Tuesday, October 25, 2022 - 11:58:11 AM

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  • HAL Id : tel-03783626, version 1

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Sofiane Arradi-Alaoui. Information auxiliaire non paramétrique : géometrie de l'information, processus empirique et applications. Théorie de l'information [cs.IT]. Université Paul Sabatier - Toulouse III, 2022. Français. ⟨NNT : 2022TOU30096⟩. ⟨tel-03783626⟩

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