Global rates of convergence of the MLEs of log-concave and s-concave densities

Charles R. Doss, Jon A. Wellner

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when -1 < s <∞ where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s <-1.

Original languageEnglish (US)
Pages (from-to)954-981
Number of pages28
JournalAnnals of Statistics
Volume44
Issue number3
DOIs
StatePublished - Jun 2016

Keywords

  • Bracketing entropy
  • Consistency
  • Empirical processes
  • Global rate
  • Hellinger metric
  • Log-concave
  • S-concave

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