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.
Publié le : 2001-12-14
Classification:
Convex function,
estimation,
Gaussian process,
integrated Brownian motion,
least squares,
62G05,
60G15,
62E20
@article{1015345957,
author = {Groeneboom, Piet and Jongbloed, Geurt and Wellner, Jon A.},
title = {A Canonical Process for Estimation of Convex Functions: The
"Invelope" of Integrated Brownian Motion $+t^4$},
journal = {Ann. Statist.},
volume = {29},
number = {2},
year = {2001},
pages = { 1620-1652},
language = {en},
url = {http://dml.mathdoc.fr/item/1015345957}
}
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., Tome 29 (2001) no. 2, pp. 1620-1652. http://gdmltest.u-ga.fr/item/1015345957/