40 research outputs found
Universal pointwise selection rule in multivariate function estimation
In this paper, we study the problem of pointwise estimation of a multivariate
function. We develop a general pointwise estimation procedure that is based on
selection of estimators from a large parameterized collection. An upper bound
on the pointwise risk is established and it is shown that the proposed
selection procedure specialized for different collections of estimators leads
to minimax and adaptive minimax estimators in various settings.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ144 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Recovering convex boundaries from blurred and noisy observations
We consider the problem of estimating convex boundaries from blurred and
noisy observations. In our model, the convolution of an intensity function
is observed with additive Gaussian white noise. The function is assumed to
have convex support whose boundary is to be recovered. Rather than directly
estimating the intensity function, we develop a procedure which is based on
estimating the support function of the set . This approach is closely
related to the method of geometric hyperplane probing, a well-known technique
in computer vision applications. We establish bounds that reveal how the
estimation accuracy depends on the ill-posedness of the convolution operator
and the behavior of the intensity function near the boundary.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Woodroofe's one-armed bandit problem revisited
We consider the one-armed bandit problem of Woodroofe [J. Amer. Statist.
Assoc. 74 (1979) 799--806], which involves sequential sampling from two
populations: one whose characteristics are known, and one which depends on an
unknown parameter and incorporates a covariate. The goal is to maximize
cumulative expected reward. We study this problem in a minimax setting, and
develop rate-optimal polices that involve suitable modifications of the myopic
rule. It is shown that the regret, as well as the rate of sampling from the
inferior population, can be finite or grow at various rates with the time
horizon of the problem, depending on "local" properties of the covariate
distribution. Proofs rely on martingale methods and information theoretic
arguments.Comment: Published in at http://dx.doi.org/10.1214/08-AAP589 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality
We address the problem of density estimation with -loss by
selection of kernel estimators. We develop a selection procedure and derive
corresponding -risk oracle inequalities. It is shown that the
proposed selection rule leads to the estimator being minimax adaptive over a
scale of the anisotropic Nikol'skii classes. The main technical tools used in
our derivations are uniform bounds on the -norms of empirical
processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011),
to appear].Comment: Published in at http://dx.doi.org/10.1214/11-AOS883 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Hough transform estimator
This article pursues a statistical study of the Hough transform, the
celebrated computer vision algorithm used to detect the presence of lines in a
noisy image. We first study asymptotic properties of the Hough transform
estimator, whose objective is to find the line that ``best'' fits a set of
planar points. In particular, we establish strong consistency and rates of
convergence, and characterize the limiting distribution of the Hough transform
estimator. While the convergence rates are seen to be slower than those found
in some standard regression methods, the Hough transform estimator is shown to
be more robust as measured by its breakdown point. We next study the Hough
transform in the context of the problem of detecting multiple lines. This is
addressed via the framework of excess mass functionals and modality testing.
Throughout, several numerical examples help illustrate various properties of
the estimator. Relations between the Hough transform and more mainstream
statistical paradigms and methods are discussed as well.Comment: Published at http://dx.doi.org/10.1214/009053604000000760 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the shape-from-moments problem and recovering edges from noisy Radon data
We consider the problem of reconstructing a planar convex set from noisy observations of its moments. An estimation method based on pointwise recovering of the support function of the set is developed. We study intrinsic accuracy limitations in the shape-from-moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error. It is shown that the proposed estimator is near-optimal in the sense of the order. An application to tomographic reconstruction is discussed, and it is indicated how the proposed estimation method can be used for recovering edges from noisy Radon data
GENERAL PROCEDURE FOR SELECTING LINEAR ESTIMATORS
International audienceIn the general statistical experiment model we propose a procedure for selecting an estimator from a given family of linear estimators. We derive an upper bound on the risk of the selected estimator and demontrate how this result can be used in order to construct minimax and adaptive minimax estimators in specic nonparametric estimation problems