Modèle de Cox : estimation par sélection de modèle et modèle de chocs bivariés

Letué, Frédérique

Directeur de thèse : Bretagnolle, Jean

Membres du jury : Van der Laan, Mark ; Gross, Shulamith T. ; Bretagnolle, Jean ; Grégoire, Gérard ; Lelièvre, Eva ; Massart, Pascal ; Pons, Odile

Thèses d'Orsay, no. 583 (2000) , 144 p.

Résumé

This thesis contains two independent parts, both based on the Cox model.

In the first chapter which presents a joint work with Gwénaëlle Castellan, the Cox model is considered when the regression function of the covariates is not necessarily linear. To estimate this regression function, we devise a nonparametric estimation by model selection. A model is defined as a $L_{\infty}$ -ball of some finite-dimensional linear space of functions. In each model, the regression function is estimated by maximizing the Cox partial log-likelihood. We define then some penalized maximum partial log-likelihood estimator, from this collection of estimators. We give a risk bound for our estimator, in comparison to the smallest risk bound over the considered estimators collection.

In the second chapter, we propose a semiparametric shock model in order to model situations in demography where the biographies of a pair of individuals cannot be considered as independent. For that purpose, we construct two dependent counting processes representing these biographies in such a way that, whenever either one of both counting processes jumps, the hazard rate of the other one is instantaneously multiplied by a constant, called a shock parameter. Moreover, these counting processes may be censored. In such a context, assuming a Cox model, we propose maximum partial log-likelihood estimators for the shock parameters and for the Cox regression parameters, from a sample of independent and identically distributed, possibly censored pairs. Consistency and asymptotic normality of these estimators are established. We illustrate our results with simulations.

Classification : 62G05, 62G07, 62M09, 62F10, 62J15, 62J15, 62P10, 62P25
Keywords: Bivariate censored data, Bivariate survival analysis, Cox model, Kullback-Leibler Information, Model selection, Nonparametric estimation, Penalization.

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     author = {Letu\'e, Fr\'ed\'erique},
     title = {Mod\`ele de {Cox} : estimation par s\'election de mod\`ele et mod\`ele de chocs bivari\'es},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud U.F.R. Scientifique d'Orsay},
     number = {583},
     year = {2000},
     language = {fr},
     url = {http://www.numdam.org/item/BJHTUP11_2000__0583__P0_0/}
}

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Letué, Frédérique. Modèle de Cox : estimation par sélection de modèle et modèle de chocs bivariés. Thèses d'Orsay, no. 583 (2000), 144 p. http://numdam.org/item/BJHTUP11_2000__0583__P0_0/

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