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December 2001 A Canonical Process for Estimation of Convex Functions: The "Invelope" of Integrated Brownian Motion $+t^4$
Piet Groeneboom, Geurt Jongbloed, Jon A. Wellner
Ann. Statist. 29(6): 1620-1652 (December 2001). DOI: 10.1214/aos/1015345957

Abstract

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the greatest convex minorant of Brownian motion plus a parabolic drift in the problem of estimating monotone functions. An iterative cubic spline algorithm is introduced that solves the constrained least squares problem in the limit situation and some results, obtained by applying this algorithm, are shown to illustrate the theory.

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Piet Groeneboom. Geurt Jongbloed. Jon A. Wellner. "A Canonical Process for Estimation of Convex Functions: The "Invelope" of Integrated Brownian Motion $+t^4$." Ann. Statist. 29 (6) 1620 - 1652, December 2001. https://doi.org/10.1214/aos/1015345957

Information

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

zbMATH: 1043.62026
MathSciNet: MR1891741
Digital Object Identifier: 10.1214/aos/1015345957

Subjects:
Primary: 62G05
Secondary: 60G15, 62E20

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 6 • December 2001
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