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Theoretical study of some statistical procedures applied to complex data

Abstract : The main part of this thesis aims at studying the theoretical and algorithmic aspects of three distinct statistical procedures. The first problem is the binary matrix completion. We propose an estimator based on a variational approximation of a pseudo-Bayesian estimator. We use a different loss function of the ones used in the literature. We are able to compute non asymptotic risk bounds. It is much faster to compute the estimator than a MCMC method and we show on examples that it is efficient in practice. In a second part we study the theoretical properties of the regularized empirical risk minimizer for Lipschitz loss functions. We are therefore able to apply it on the logistic regression with the SLOPE regularization and on the matrix completion as well. The third chapter develops an Expectation-Propagation approximation when the likelihood is not explicit. We then use an ABC approximation in a second stage. This procedure may be applied to many models and is more precise and faster than the classic ABC approximation. It is used in a spatial extremes model.
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Submitted on : Friday, December 15, 2017 - 4:46:02 PM
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Vincent R. Cottet. Theoretical study of some statistical procedures applied to complex data. Statistics [math.ST]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLG002⟩. ⟨tel-01665304⟩

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