Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

Details Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

2007-09-04

arxiv.org/abs/0709.0334v4

Bernoulli 2009, Vol. 15, No. 1, 40-68

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/08-BEJ141 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ141

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 $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.

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