Confidence regions, non-parametric regression
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Date
2007-05-25T12:06:48Z
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Abstract
In this paper we offer a unified approach to the problem of nonparametric
regression on the unit interval. It is based on a universal,
honest and non-asymptotic confidence region An which is defined by a
set of linear inequalities involving the values of the functions at the design
points. Interest will typically centre on certain simplest functions
in An where simplicity can be defined in terms of shape (number of local
extremes, intervals of convexity/concavity) or smoothness (bounds
on derivatives) or a combination of both. Once some form of regularization
has been decided upon the confidence region can be used
to provide honest non-asymptotic confidence bounds which are less
informative but conceptually much simpler. Although the procedure
makes no attempt to minimize any loss function such as MISE the
resulting estimates have optimal rates of convergence in the supremum
norm both for shape and smoothness regularization. We show
that rates of convergence can be misleading even for samples of size
n = 10^6 and propose a different form of asymptotics which allows
model complexity to increase with sample size.
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Keywords
Confidence bounds, Nonparametric regression, Shape regularization