The Annals of Statistics

Approximation by log-concave distributions, with applications to regression

Lutz Dümbgen, Richard Samworth, and Dominic Schuhmacher

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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.

Article information

Ann. Statist., Volume 39, Number 2 (2011), 702-730.

First available in Project Euclid: 9 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E17: Approximations to distributions (nonasymptotic) 62G05: Estimation 62G07: Density estimation 62G08: Nonparametric regression 62G35: Robustness 62H12: Estimation

Convex support isotonic regression linear regression Mallows distance projection weak semicontinuity


Dümbgen, Lutz; Samworth, Richard; Schuhmacher, Dominic. Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 (2011), no. 2, 702--730. doi:10.1214/10-AOS853.

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