Estimation par Minimum de Contraste Régulier et Heuristique de Pente en Sélection de Modèles

Abstract : This thesis is devoted to the theoritical analysis of a method of calibration of penalties for model selection procedures. This method is based on a heuristics, called the "slope heuristics", which stipulate the existence of a minimal penalty such that the optimal solution of the penalisation problem is twice this penalty. In practice, one estimate the optimal penalty by the previous estimation of the minimal one, characterized by a complete changing in the behavior of the model selection procedure occuring around the minimal level of penalty. The theoritical understanding of the slope phenomenon is based on a sharp control with exact constant of the deviations of the excess risk and the empirical excess risk of the considered estimators, respectively measuring their performance in prediction and their empirical performance. This suggests a strong specification of the structure of the tackled problem. We validate the slope heuristics in a general framework based on a new notion in M-estimation, that we call "regular contrast", and we develop an original methodology of proof, allowing the simultaneous treatment of the problems of upper and lower bounds for the excess risk. We thus recover most of the known results concerning the slope phenomenon. Indeed, we give tree examples of regular contrast estimation, namely least-squares regression on linear models, least-squares density estimation on affine models and maximum likelihood density estimation on convex sets. This permits us to extend the previously known results in the regression setting to more general linear models and to validate the slope heuristics for a non-quadratic risk considering the maximum likelihood estimation of density. Finally, our methodology of proof provides with precise directions of research for non-regular situations, as one can find in classification and more generally in the statistical learning theory.
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https://tel.archives-ouvertes.fr/tel-00569372
Contributor : Adrien Saumard <>
Submitted on : Thursday, February 24, 2011 - 8:20:20 PM
Last modification on : Friday, November 16, 2018 - 1:27:59 AM
Long-term archiving on: Tuesday, November 6, 2012 - 3:00:23 PM

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Adrien Saumard. Estimation par Minimum de Contraste Régulier et Heuristique de Pente en Sélection de Modèles. Mathématiques [math]. Université Rennes 1, 2010. Français. ⟨tel-00569372⟩

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