## The Annals of Statistics

### A Canonical Process for Estimation of Convex Functions: The "Invelope" of Integrated Brownian Motion $+t^4$

#### 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.

#### Article information

Source
Ann. Statist., Volume 29, Number 6 (2001), 1620-1652.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015345957

Digital Object Identifier
doi:10.1214/aos/1015345957

Mathematical Reviews number (MathSciNet)
MR1891741

Zentralblatt MATH identifier
1043.62026

Subjects
Primary: 62G05: Estimation
Secondary: 60G15: Gaussian processes 62E20: Asymptotic distribution theory

#### Citation

Groeneboom, Piet; Jongbloed, Geurt; Wellner, Jon A. A Canonical Process for Estimation of Convex Functions: The "Invelope" of Integrated Brownian Motion $+t^4$. Ann. Statist. 29 (2001), no. 6, 1620--1652. doi:10.1214/aos/1015345957. https://projecteuclid.org/euclid.aos/1015345957

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