Year: 2006, Volume: 7, Issue: 34, Pages: 971−981
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-Leibler divergence (KL), the Hellinger's distance and the L1-distance. The lower bound, with respect to the KL distance, can be achieved by the on-line type estimate suggested, among others, by Yang (2000a). 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.