• Bernoulli
  • Volume 23, Number 1 (2017), 219-248.

Optimal exponential bounds for aggregation of density estimators

Pierre C. Bellec

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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.

Article information

Bernoulli, Volume 23, Number 1 (2017), 219-248.

Received: January 2015
Revised: April 2015
First available in Project Euclid: 27 September 2016

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aggregation concentration inequality density estimation minimax lower bounds minimax optimality model selection sharp oracle inequality


Bellec, Pierre C. Optimal exponential bounds for aggregation of density estimators. Bernoulli 23 (2017), no. 1, 219--248. doi:10.3150/15-BEJ742.

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