Optimal exponential bounds for aggregation of density estimators

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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.

Original languageEnglish (US)
Pages (from-to)219-248
Number of pages30
JournalBernoulli
Volume23
Issue number1
DOIs
StatePublished - Feb 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Keywords

  • Aggregation
  • Concentration inequality
  • Density estimation
  • Minimax lower bounds
  • Minimax optimality
  • Model selection
  • Sharp oracle inequality

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