  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.
