We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
Publié le : 2009-06-15
Classification:
Asymptotic distribution,
integral of Brownian motion,
invelope process,
log-concave density estimation,
lower bounds,
maximum likelihood,
mode estimation,
nonparametric estimation,
qualitative assumptions,
shape constraints,
strongly unimodal,
unimodal,
62N01,
62G20,
62G05
@article{1239369023,
author = {Balabdaoui, Fadoua and Rufibach, Kaspar and Wellner, Jon A.},
title = {Limit distribution theory for maximum likelihood estimation of a log-concave density},
journal = {Ann. Statist.},
volume = {37},
number = {1},
year = {2009},
pages = { 1299-1331},
language = {en},
url = {http://dml.mathdoc.fr/item/1239369023}
}
Balabdaoui, Fadoua; Rufibach, Kaspar; Wellner, Jon A. Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist., Tome 37 (2009) no. 1, pp. 1299-1331. http://gdmltest.u-ga.fr/item/1239369023/