Open Access
February 2017 Optimal exponential bounds for aggregation of density estimators
Pierre C. Bellec
Bernoulli 23(1): 219-248 (February 2017). DOI: 10.3150/15-BEJ742


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.


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Pierre C. Bellec. "Optimal exponential bounds for aggregation of density estimators." Bernoulli 23 (1) 219 - 248, February 2017.


Received: 1 January 2015; Revised: 1 April 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1368.62085
MathSciNet: MR3556772
Digital Object Identifier: 10.3150/15-BEJ742

Keywords: Aggregation , concentration inequality , Density estimation , minimax lower bounds , Minimax optimality , Model selection , Sharp oracle inequality

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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