A canonical process for estimation of convex functions

The "invelope" of integrated Brownian motion +t4

Journal article (2001)
Department
Delft Institute of Applied Mathematics (EEMCS) (TU Delft)
Copyright: © 2001 Institute of Mathematical Statistics
DOI: https://doi.org/doi:10.1214/aos/1015345957
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Published Date

01-12-2001

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Source:

Annals of Statistics, 29 (6), 2001

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

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.