Linear and convex aggregation of density estimators.
Philippe Rigollet and Alexandre Tsybakov
Mathematical Methods of Statistics
We study the problem of learning the best linear and convex
combination of $M$ estimators of a density with respect to the mean
squared risk. We suggest aggregation procedures and we prove sharp
oracle inequalities for their risks, i.e., oracle inequalities with
leading constant 1. We also obtain lower bounds showing that these
procedures attain optimal rates of aggregation. As an example, we
consider aggregation of multivariate kernel density estimators with
different bandwidths. We show that linear and convex aggregates
mimic the kernel oracles in asymptotically exact sense. 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.