  We study the problem of linear and convex aggregation of $M$ estimators of a
density with respect to the mean squared risk. We provide procedures for linear
and convex aggregation and we prove oracle inequalities for their risks. We
also obtain lower bounds showing that these procedures are rate optimal in a
minimax sense. As an example, we apply general results to aggregation of
multivariate kernel density estimators with different bandwidths. We show that
linear and convex aggregates mimic the kernel oracles in asymptotically exact
sense for a large class of kernels including Gaussian, Silverman's and
Pinsker's ones. We prove that, for Pinsker's kernel, the proposed aggregates
are sharp asymptotically minimax simultaneously over a large scale of Sobolev
classes of densities. Finally, we provide simulations demonstrating performance
of the convex aggregation procedure.
