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Approximation by log-concave distributions, with applications to regression. (English) Zbl 1216.62023

Summary: 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 C. L. Mallows distance \(D_{1}(\cdot , \cdot )\) [Ann. Math. Stat. 43, 508–515 (1972; Zbl 0238.60017)]. 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 \(Y=\mu (X)+\varepsilon \), where \(X\) and \(\varepsilon \) are independent, \(\mu (\cdot )\) belongs to a certain class of regression functions while \(\varepsilon \) is a random error with log-concave density and mean zero.

MSC:

62E17 Approximations to statistical distributions (nonasymptotic)
62G07 Density estimation
62G08 Nonparametric regression and quantile regression
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models

Citations:

Zbl 0238.60017
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References:

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