Lower bounds and aggregation in density estimation

Guillaume Lecué (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

In this paper we prove the optimality of an aggregation procedure. We prove lower bounds for aggregation of model selection type of $M$ density estimators for the Kullback-Leiber divergence (KL), the Hellinger's distance and the $L_1$-distance. The lower bound, with respect to the KL distance, can be achieved by the on-line type estimate suggested, among others, by Yang (2000). Combining these results, we state that $\log M/n$ is an optimal rate of aggregation in the sense of Tsybakov (2003), where $n$ is the sample size.

Mots clés: Aggregation; optimal rates; Kullback-Leiber divergence

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Guillaume Lecué. Lower bounds and aggregation in density estimation. 2006. ⟨hal-00021232⟩