5,460 research outputs found
On adaptive wavelet estimation of a class of weighted densities
We investigate the estimation of a weighted density taking the form
, where denotes an unknown density, the associated
distribution function and is a known (non-negative) weight. Such a class
encompasses many examples, including those arising in order statistics or when
is related to the maximum or the minimum of (random or fixed)
independent and identically distributed (\iid) random variables. We here
construct a new adaptive non-parametric estimator for based on a plug-in
approach and the wavelets methodology. For a wide class of models, we prove
that it attains fast rates of convergence under the risk with
(not only for corresponding to the mean integrated squared
error) over Besov balls. The theoretical findings are illustrated through
several simulations
Confidence bands in density estimation
Given a sample from some unknown continuous density
, we construct adaptive confidence bands that are
honest for all densities in a "generic" subset of the union of -H\"older
balls, , where is a fixed but arbitrary integer. The exceptional
("nongeneric") set of densities for which our results do not hold is shown to
be nowhere dense in the relevant H\"older-norm topologies. In the course of the
proofs we also obtain limit theorems for maxima of linear wavelet and kernel
density estimators, which are of independent interest.Comment: Published in at http://dx.doi.org/10.1214/09-AOS738 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
MDL Denoising Revisited
We refine and extend an earlier MDL denoising criterion for wavelet-based
denoising. We start by showing that the denoising problem can be reformulated
as a clustering problem, where the goal is to obtain separate clusters for
informative and non-informative wavelet coefficients, respectively. This
suggests two refinements, adding a code-length for the model index, and
extending the model in order to account for subband-dependent coefficient
distributions. A third refinement is derivation of soft thresholding inspired
by predictive universal coding with weighted mixtures. We propose a practical
method incorporating all three refinements, which is shown to achieve good
performance and robustness in denoising both artificial and natural signals.Comment: Submitted to IEEE Transactions on Information Theory, June 200
A note on an Adaptive Goodness-of-Fit test with Finite Sample Validity for Random Design Regression Models
Given an i.i.d. sample from the random
design regression model with , in this paper we consider the problem of testing the (simple) null
hypothesis , against the alternative for a fixed , where denotes the marginal distribution of the
design variable . The procedure proposed is an adaptation to the regression
setting of a multiple testing technique introduced by Fromont and Laurent
(2005), and it amounts to consider a suitable collection of unbiased estimators
of the --distance ,
rejecting the null hypothesis when at least one of them is greater than its
quantile, with calibrated to obtain a level--
test. To build these estimators, we will use the warped wavelet basis
introduced by Picard and Kerkyacharian (2004). We do not assume that the errors
are normally distributed, and we do not assume that and are
independent but, mainly for technical reasons, we will assume, as in most part
of the current literature in learning theory, that is uniformly
bounded (almost everywhere). We show that our test is adaptive over a
particular collection of approximation spaces linked to the classical Besov
spaces
Adaptive circular deconvolution by model selection under unknown error distribution
We consider a circular deconvolution problem, in which the density of a
circular random variable must be estimated nonparametrically based on an
i.i.d. sample from a noisy observation of . The additive measurement
error is supposed to be independent of . The objective of this work was to
construct a fully data-driven estimation procedure when the error density
is unknown. We assume that in addition to the i.i.d. sample from ,
we have at our disposal an additional i.i.d. sample drawn independently from
the error distribution. We first develop a minimax theory in terms of both
sample sizes. We propose an orthogonal series estimator attaining the minimax
rates but requiring optimal choice of a dimension parameter depending on
certain characteristics of and , which are not known in practice.
The main issue addressed in this work is the adaptive choice of this dimension
parameter using a model selection approach. In a first step, we develop a
penalized minimum contrast estimator assuming that the error density is known.
We show that this partially adaptive estimator can attain the lower risk bound
up to a constant in both sample sizes and . Finally, by randomizing the
penalty and the collection of models, we modify the estimator such that it no
longer requires any previous knowledge of the error distribution. Even when
dispensing with any hypotheses on , this fully data-driven estimator
still preserves minimax optimality in almost the same cases as the partially
adaptive estimator. We illustrate our results by computing minimal rates under
classical smoothness assumptions.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ422 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
Exact oracle inequality for a sharp adaptive kernel density estimator
In one-dimensional density estimation on i.i.d. observations we suggest an
adaptive cross-validation technique for the selection of a kernel estimator.
This estimator is both asymptotic MISE-efficient with respect to the monotone
oracle, and sharp minimax-adaptive over the whole scale of Sobolev spaces with
smoothness index greater than 1/2. The proof of the central concentration
inequality avoids "chaining" and relies on an additive decomposition of the
empirical processes involved
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