Grenander functionals and Cauchy's formula

Piet Groeneboom*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)
50 Downloads (Pure)

Abstract

Let (Formula presented.) be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left-continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L2-distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L2-distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.

Original languageEnglish
Pages (from-to)275-294
Number of pages20
JournalScandinavian Journal of Statistics
Volume48
Issue number1
DOIs
Publication statusPublished - 2020

Keywords

  • Cauchy's formula
  • Grenander estimator
  • integral statistics
  • saddle points

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