Approximation by log-concave distributions, with applications to regression

Details Approximation by log-concave distributions, with applications to regression Approximation by log-concave distributions, with applications to regression

2010-02-18

arxiv.org/abs/1002.3448v4

Annals of Statistics 2011, Vol. 39, No. 2, 702-730

Mathematics

Statistics Theory

Version 3 is the technical report cited in the published paper. Published in at http://dx.doi.org/10.1214/10-AOS853 the Annals

Scientific paper

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 $D_1(\cdot,\cdot)$. 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.

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