We study the nonparametric estimation of univariate monotone and
unimodal densities usingthe maximum smoothed likelihood approach. The monotone
estimator is the derivative of the least concave majorant of the distribution
correspondingto a kernel estimator.We prove that the mapping on distributions
$\Phi$ with density $\varphi$,
$$\varphi \mapsto \text{the derivative of the
least concave majorant of $\Phi},$$
¶ is a contraction in all $L^P$ norms $(1 \leq p \leq \infty)$, and
some other “distances” such as the Hellinger and
Kullback–Leibler distances. The contractivity implies error bounds for
monotone density estimation. Almost the same error bounds hold for unimodal
estimation.
Publié le : 2000-05-14
Classification:
Maximum likelihood estimation,
monotone and unimodal densities,
least concave majorants,
contractions,
$L^1$ error bounds,
62G07
@article{1015952005,
author = {Eggermont, P. P. B. and LaRiccia, V. N.},
title = {Maximum likelihood estimation of smooth monotone and unimodal
densities},
journal = {Ann. Statist.},
volume = {28},
number = {3},
year = {2000},
pages = { 922-947},
language = {en},
url = {http://dml.mathdoc.fr/item/1015952005}
}
Eggermont, P. P. B.; LaRiccia, V. N. Maximum likelihood estimation of smooth monotone and unimodal
densities. Ann. Statist., Tome 28 (2000) no. 3, pp. 922-947. http://gdmltest.u-ga.fr/item/1015952005/