Grenander functionals and Cauchy's formula

Abstract

Letfn be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this in [6] 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 L 2 -distance of the Grenander estimator to the uniform density was derived in [12] by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in [8] to prove a central limit result of Sparre Andersen [19] on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result in [12] and also prove a similar asymptotic normality results for the entropy functional. In [12] the limit distribution of the sums of gamma and Poisson variables on which the conditioning was done did not have the right form, which is corrected here. Cauchy's formula and saddle point methods are the main tools in our development.

On efficiency of the plug-in principle for estimating smooth integrated functionals of a nonincreasing density

“…was recently pointed out inGroeneboom (2019).imsart-generic ver. 2014/10/16 file: Eff-Gren-Arxiv-III.tex date: April 16, 2019…”

“…The following lemma, crucial to our proof, is the analogue of Groeneboom and Pyke (1983, Lemma 3.1); a correction to the characteristic function formula in Groeneboom and Pyke (1983, Lemma 3.1) was recently pointed out in Groeneboom (2019).…”