## Bernoulli

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

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 15, Number 1 (2009), 40-68.

Dates
First available in Project Euclid: 3 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1233669882

Digital Object Identifier
doi:10.3150/08-BEJ141

Mathematical Reviews number (MathSciNet)
MR2546798

Zentralblatt MATH identifier
1200.62030

#### Citation

Dümbgen, Lutz; Rufibach, Kaspar. Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli 15 (2009), no. 1, 40--68. doi:10.3150/08-BEJ141. https://projecteuclid.org/euclid.bj/1233669882

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