Adaptive estimation of the conditional intensity of marker-dependent counting processes

Comte, F.; Gaïffas, S.; Guilloux, A.

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1171-1196.

Dans ce travail, nous proposons un estimateur original de l'intensité conditionnelle d'un processus de comptage marqué, c'est-à-dire d'un processus de comptage dépendant de covariables. Nous utilisons une méthode de sélection de modèle et nous obtenons pour notre estimateur, une borne non asymptotique du risque quadratique sur un compact. Nous vérifions ensuite que l'estimateur atteint automatiquement une vitesse de convergence sur des classes fonctionnelles de régularité anisotropique fixée mais inconnue. Enfin, nous démontrons une borne inférieure qui garantit l'optimalité de la vitesse obtenue. Une brève illustration de la façon dont fonctionne l'estimateur dans le contexte de l'estimation du taux de risque instantané conditionnel est fournie pour conclure.

We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation.

Classification : 62N02, 62G05
Mots clés : marker-dependent counting process, conditional intensity, model selection, adaptive estimation, minimax and nonparametric methods, censored data, conditional hazard function

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