2010/02/28 by Lutz Dümbgen, Lutz Duembgen, Richard Samworth +1
Mathematics · #Approximation theory #Class (philosophy) #Consistency (knowledge bases) #Estimator #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Regression #Regression analysis #Space (punctuation) #Statistical Methods and Inference #Strong consistency #Weak consistency #math.PR #math.ST #stat.ME #stat.TH
paper · pdf · doi:10.1214/10-aos853
published as Annals of Statistics 2011, Vol. 39, No. 2, 702-730 · 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)
openalex publication_date 2011/03/09 · arxiv created 2011/05/11 · arxiv updated 2011/10/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.