The Annals of Statistics

Estimation of a Convex Function: Characterizations and Asymptotic Theory

Piet Groeneboom, Geurt Jongbloed, and Jon A. Wellner

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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.

Article information

Ann. Statist., Volume 29, Number 6 (2001), 1653-1698.

First available in Project Euclid: 5 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62E20: Asymptotic distribution theory

Convex dinsity estimation regression function maximum likelihood least squares integrated Brownian motion


Groeneboom, Piet; Jongbloed, Geurt; Wellner, Jon A. Estimation of a Convex Function: Characterizations and Asymptotic Theory. Ann. Statist. 29 (2001), no. 6, 1653--1698. doi:10.1214/aos/1015345958.

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