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. 1991 Mathematics Subject Classification. Primary 62G08, Secondary 62C20, 62G05, 62G20.
Linear and convex aggregation of densities, based on an L 2 criterion, are studied in REF ,
Linear and convex aggregation of density estimators