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April 2011 Approximation by log-concave distributions, with applications to regression
Lutz Dümbgen, Richard Samworth, Dominic Schuhmacher
Ann. Statist. 39(2): 702-730 (April 2011). DOI: 10.1214/10-AOS853


We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback–Leibler-type functional. We show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows distance D1(⋅, ⋅). This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response Y=μ(X)+ε, where X and ε are independent, μ(⋅) belongs to a certain class of regression functions while ε is a random error with log-concave density and mean zero.


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Lutz Dümbgen. Richard Samworth. Dominic Schuhmacher. "Approximation by log-concave distributions, with applications to regression." Ann. Statist. 39 (2) 702 - 730, April 2011.


Published: April 2011
First available in Project Euclid: 9 March 2011

zbMATH: 1216.62023
MathSciNet: MR2816336
Digital Object Identifier: 10.1214/10-AOS853

Primary: 62E17 , 62G05 , 62G07 , 62G08 , 62G35 , 62H12

Keywords: convex support , isotonic regression , Linear regression , Mallows distance , projection , weak semicontinuity

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 2 • April 2011
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