On adaptive minimax density estimation on Rd

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We address the problem of adaptive minimax density estimation on Rd with Lp-loss on the anisotropic Nikol’skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in order over the complete scale of the anisotropic Nikol’skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.

Original languageEnglish
Pages (from-to)479-543
Number of pages65
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Aug 2014

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2013.


  • Adaptive estimation
  • Density estimation
  • Kernel estimators
  • Lp-risk
  • Oracle inequality

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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