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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Abstract

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

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

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015345958

Digital Object Identifier
doi:10.1214/aos/1015345958

Mathematical Reviews number (MathSciNet)
MR1891742

Zentralblatt MATH identifier
1043.62027

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1015345958


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