Nonparametric density estimation under unimodality and monotonicity constraints

Ming Yen Cheng*, Theo Gasser, Peter Hall

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

20 Citations (Scopus)


We introduce a recursive method for estimating a probability density subject to constraints of unimodality or monotonicity. It uses an empirical estimate of the probability transform to construct a sequence of maps of a known template, which satisfies the constraints. The algorithm may be employed without a smoothing step, in which case it produces step-function approximations to the sampling density. More satisfactorily, a certain amount of smoothing may be interleaved between each recursion, in which case the estimate is smooth. The amount of smoothing may be chosen using a standard cross-validation algorithm. Unlike other methods for density estimation, however, the recursive approach is robust against variation of the amount of smoothing, and so choice of bandwidth is not critical.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalJournal of Computational and Graphical Statistics
Issue number1
Publication statusPublished - Mar 1999

User-Defined Keywords

  • Curve estimation
  • Isotonic regression
  • Iteration
  • Kernel methods
  • Mode
  • Mode testing
  • Probability transform
  • Recursion
  • Smoothing
  • Turning point


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