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


Dümbgen, L; Rufibach, K (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli, 15(1):40-68.

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 op(n−1/2) under certain regularity assumptions.

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 op(n−1/2) under certain regularity assumptions.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:04 Faculty of Medicine > Epidemiology, Biostatistics and Prevention Institute (EBPI)
Dewey Decimal Classification:610 Medicine & health
Language:English
Date:2009
Deposited On:07 Apr 2009 13:40
Last Modified:05 Apr 2016 13:04
Publisher:Bernoulli Society for Mathematical Statistics and Probability
ISSN:1350-7265
Additional Information:2009 © Bernoulli Society for Mathematical Statistics and Probability
Publisher DOI:https://doi.org/10.3150/08-BEJ141
Official URL:http://dx.doi.org/10.3150/08-BEJ141

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