Some applications of concentration inequalities to statistics

Nous presentons quelques applications d'inegalites de concentration a la resolution de problemes de selection de modeles en statistique. Nous etudions en detail deux exemples pour lesquels cette approche s'avere particulirement fructueuse. Nous considerons tout d'abord le classique mais delicat probleme du choix d'un bon histogramme. Nous presentons un extrait de travail de Castellan sur la question, mettant en evidence que la structure meme des inegalites de concentration de Talagrand pour des processus empiriques influence directement la construction d'un critere de selection de type Akaike modifie. Nous presentons egalement un nouveau theoreme de selection de modeles bien adapte a la resolution de problemes d'apprentissage. Ce resultat permet de reinterpreter et d'ameliorer la methode dite de minimisation structurelle du risque due a Vapnik.

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