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

Lutz Dümbgen and Kaspar Rufibach

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

First available in Project Euclid: 3 February 2009

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adaptivity bracketing exponential inequality gap problem hazard function method of caricatures shape constraints


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.

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