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Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
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Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
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Journal Article
Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds
2012
Overview
Let
X
1
, . . . ,
X
n
be a random sample from some unknown probability density
f
defined on a compact homogeneous manifold
M
of dimension
d
≥ 1. Consider a ‘needlet frame’
describing a localised projection onto the space of eigenfunctions of the Laplace operator on
M
with corresponding eigenvalues less than 2
2
j
, as constructed in Geller and Pesenson (J Geom Anal
2011
). We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator
f
n
(
j
) obtained from an empirical estimate of the needlet projection
of
f
. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density
f
. The confidence bands are adaptive over classes of differentiable and Hölder-continuous functions on
M
that attain their Hölder exponents.
Publisher
Springer-Verlag,Springer Nature B.V,Springer Verlag
