Mixing Strategies for Density Estimation

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Abstract

General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and squared L2 losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimax rate adaptive estimator exists for a given countable collection of density classes; that is, a single estimator can be constructed to be simultaneously minim ax-rate optimal for all the function classes being considered. A demonstration is given for high-dimensional density estimation on [0, 1]d where the constructed estimator adapts to smoothness and interaction-order over some piecewise Besov classes and is consistent for all the densities with finite entropy.

Original languageEnglish (US)
Pages (from-to)75-87
Number of pages13
JournalAnnals of Statistics
Volume28
Issue number1
DOIs
StatePublished - Feb 2000

Keywords

  • Adaptation with respect to estimation strategies
  • Density estimation
  • Minimax adaptation
  • Rates of convergence

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