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December 2001 Estimation of a Convex Function: Characterizations and Asymptotic Theory
Piet Groeneboom, Geurt Jongbloed, Jon A. Wellner
Ann. Statist. 29(6): 1653-1698 (December 2001). DOI: 10.1214/aos/1015345958


We study nonparametric estimation of convexregression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case).We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an “invelope function” for integrated two-sided Brownian motion $+t^4$ which is established in a companion paper by Groeneboom, Jongbloed and Wellner.


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Piet Groeneboom. Geurt Jongbloed. Jon A. Wellner. "Estimation of a Convex Function: Characterizations and Asymptotic Theory." Ann. Statist. 29 (6) 1653 - 1698, December 2001.


Published: December 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1043.62027
MathSciNet: MR1891742
Digital Object Identifier: 10.1214/aos/1015345958

Primary: 62G05
Secondary: 62E20 , 62G07 , 62G08

Keywords: convex , dinsity estimation , Integrated Brownian motion , least squares , maximum likelihood , regression function

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 6 • December 2001
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