Approximation by log-concave distributions, with applications to regression
Lutz Dümbgen, Richard Samworth, Dominic Schuhmacher
The Annals of Statistics · 2011 · Volume 39 · Issue 2
Abstract
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.
Keywords
62E17, 62G05, 62G07, 62G08, 62G35, 62H12, convex support, isotonic regression, Linear regression, Mallows distance, projection, weak semicontinuity
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