We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n)1/3 and typically (log(n)/n)2/5, whereas the difference between the empirical and estimated distribution function vanishes with rate op(n−1/2) under certain regularity assumptions.
Publié le : 2009-02-15
Classification:
adaptivity,
bracketing,
exponential inequality,
gap problem,
hazard function,
method of caricatures,
shape constraints
@article{1233669882,
author = {D\"umbgen, Lutz and Rufibach, Kaspar},
title = {Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency},
journal = {Bernoulli},
volume = {15},
number = {1},
year = {2009},
pages = { 40-68},
language = {en},
url = {http://dml.mathdoc.fr/item/1233669882}
}
Dümbgen, Lutz; Rufibach, Kaspar. Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli, Tome 15 (2009) no. 1, pp. 40-68. http://gdmltest.u-ga.fr/item/1233669882/