Aggregation of density estimators and dimension reduction

We consider the problem of model-selection-type aggregation of arbitrary density estimators using MISE risk. Given a collection of arbitrary density estimators, we propose a data-based selector of the best estimator in the collection and prove a general ready-to-use oracle inequality for the selected aggregate estimator. We then apply this inequality to the adaptive estimation of a multivariate density in a “multiple index” model. We show that the proposed aggregate estimator adapts to the unknown index space of unknown dimension in the sense that it allows us to estimate the density with the optimal rate attainable when the index space is known.