QUASI-concave density estimation

Roger Koenker, Ivan Mizera

Research output: Contribution to journalArticlepeer-review


Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.

Original languageEnglish (US)
Pages (from-to)2998-3027
Number of pages30
JournalAnnals of Statistics
Issue number5
StatePublished - Oct 2010


  • Convex optimization
  • Density estimation
  • Duality
  • Entropy
  • Semidefinite programming.
  • Shape constraints
  • Strongly unimodal
  • Unimodal

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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