Model selection for Poisson processes

Lucien Birgé (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Our purpose in this paper is to apply the general methodology for model selection based on T-estimators developed in Birgé (Model selection via testing : an alternative to (penalized) maximum likelihood estimators, Ann. Inst. Henri Poincaré Probab. et Statist., vol. 42, 273-325, 2006) to the particular situation of the estimation of the unknown mean measure of a Poisson process. We introduce a Hellinger type distance between finite positive measures to serve as our loss function and we build suitable tests between balls (with respect to this distance) in the set of mean measures. As a consequence of the existence of such tests, given a suitable family of approximating models, we can build T-estimators for the mean measure based on this family of models and analyze their performances. We provide a number of applications to adaptive intensity estimation when the square root of the intensity belongs to various smoothness classes. We also give a method for aggregation of preliminary estimators.

Mots clés: Poisson processes; intensity estimation; model selection; adaptive estimation; aggregation; robust tests

Domaines: Statistiques [math.ST]

Lucien Birgé. Model selection for Poisson processes. 2006. ⟨hal-00104631⟩