The Annals of Statistics
- Ann. Statist.
- Volume 29, Number 6 (2001), 1653-1698.
Estimation of a Convex Function: Characterizations and Asymptotic Theory
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.
Ann. Statist., Volume 29, Number 6 (2001), 1653-1698.
First available in Project Euclid: 5 March 2002
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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