Approximation by log-concave distributions, with applications to regression

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 $D_1(\cdot ,\cdot )$. 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 $Y=\mu (X)+\epsilon$, where $X$ and $\epsilon$ are independent, $\mu (\cdot )$ belongs to a certain class of regression functions while $\epsilon$ is a random error with log-concave density and mean zero.

Version 3 is the technical report cited in the published paper. Published in at http://dx.doi.org/10.1214/10-AOS853 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Approximation by log-concave distributions, with applications to regression
• Inference and Modeling with Log-concave Distributions
• Nonparametric estimation of multivariate convex-transformed densities
• Quasi-concave density estimation

Cited by in corpus (9)

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• A Bayesian nonparametric approach to log-concave density estimation
• On Convex Least Squares Estimation when the Truth is Linear
• Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its Consistency
• Maximum-Likelihood Estimation of a Log-Concave Density based on Censored Data
• Exact Solutions in Log-Concave Maximum Likelihood Estimation