APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION
By L. Duembgen, Richard J. Samworth, Dominic Schuhmacher. 2010
90 citations recorded by the source corpus.
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 (·, ·). 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) + e, where X and e are independent, μ(·) belongs to a certain class of regression functions while e is a random error with log-concave density and mean zero.
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/8992e97fcda6cf3c66e6a823623c48be842f1922
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In Applied Mathematics
In Statistics and Probability
- Yining Chen
- Richard J. Samworth
- Arlene K. H. Kim
- Bodhisattva Sen
- John D. Lafferty
- Minhua Chen
- Min Xu
- David B. Dunson
Citations in the corpus
- Generalized additive and index models with shape constraints (2014, methodology): “Possible strategies include the differential evolution method (Price et al., 2005; Dümbgen et al., 2011) or a stochastic search strategy (Dümbgen et al., 2013) described below.”
- Log-Concavity and Strong Log-Concavity: a review (2014, unknown): “We show how preservation of log-concavity and strongly log-concavity on R under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013).”
- Covering Numbers for Convex Functions (2012, unknown): “The paper is organized as follows.”