9,125 research outputs found
Honest adaptive confidence bands and self-similar functions
Confidence bands are confidence sets for an unknown function f, containing
all functions within some sup-norm distance of an estimator. In the density
estimation, regression, and white noise models, we consider the problem of
constructing adaptive confidence bands, whose width contracts at an optimal
rate over a range of H\"older classes.
While adaptive estimators exist, in general adaptive confidence bands do not,
and to proceed we must place further conditions on f. We discuss previous
approaches to this issue, and show it is necessary to restrict f to
fundamentally smaller classes of functions.
We then consider the self-similar functions, whose H\"older norm is similar
at large and small scales. We show that such functions may be considered
typical functions of a given H\"older class, and that the assumption of
self-similarity is both necessary and sufficient for the construction of
adaptive bands. Finally, we show that this assumption allows us to resolve the
problem of undersmoothing, creating bands which are honest simultaneously for
functions of any H\"older norm
Spatially-adaptive sensing in nonparametric regression
While adaptive sensing has provided improved rates of convergence in sparse
regression and classification, results in nonparametric regression have so far
been restricted to quite specific classes of functions. In this paper, we
describe an adaptive-sensing algorithm which is applicable to general
nonparametric-regression problems. The algorithm is spatially adaptive, and
achieves improved rates of convergence over spatially inhomogeneous functions.
Over standard function classes, it likewise retains the spatial adaptivity
properties of a uniform design
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