Optimal exponenetial bounds for aggregation of density estimators

Pierre Bellec (pierre.bellec@ensae.fr)
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Pierre Bellec: CREST, ENSAE, UMR CNRS 9194

No 2015-05, Working Papers from Center for Research in Economics and Statistics

Abstract: We consider the problem of model selection type aggregation in the context of density estimation. We first show that empirical risk minimization is sub-optimal for this problem and it shares this property with the exponential weights aggregate, empirical risk minimization over the convex hull of the dictionary functions, and all selectors. Using a penalty inspired by recent works on the Q-aggregation procedure, we derive a sharp oracle inequality in deviation under a simple boundedness assumption and we show that the rate is optimal in a minimax sense. Unlike the procedures based on exponential weights, this estimator is fully adaptive under the uniform prior. In particular, its construction does not rely on the sup-norm of the unknown density. By providing lower bounds with exponential tails, we show that the deviation term appearing in the sharp oracle inequalities cannot be improved.

Pages: 30
Date: 2015-06
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