15,647 research outputs found
Adaptive Hausdorff estimation of density level sets
Consider the problem of estimating the -level set
of an unknown -dimensional density
function based on independent observations from the
density. This problem has been addressed under global error criteria related to
the symmetric set difference. However, in certain applications a spatially
uniform mode of convergence is desirable to ensure that the estimated set is
close to the target set everywhere. The Hausdorff error criterion provides this
degree of uniformity and, hence, is more appropriate in such situations. It is
known that the minimax optimal rate of error convergence for the Hausdorff
metric is for level sets with boundaries that
have a Lipschitz functional form, where the parameter characterizes
the regularity of the density around the level of interest. However, the
estimators proposed in previous work are nonadaptive to the density regularity
and require knowledge of the parameter . Furthermore, previously
developed estimators achieve the minimax optimal rate for rather restricted
classes of sets (e.g., the boundary fragment and star-shaped sets) that
effectively reduce the set estimation problem to a function estimation problem.
This characterization precludes level sets with multiple connected components,
which are fundamental to many applications. This paper presents a fully
data-driven procedure that is adaptive to unknown regularity conditions and
achieves near minimax optimal Hausdorff error control for a class of density
level sets with very general shapes and multiple connected components.Comment: Published in at http://dx.doi.org/10.1214/08-AOS661 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimal rates for plug-in estimators of density level sets
In the context of density level set estimation, we study the convergence of
general plug-in methods under two main assumptions on the density for a given
level . More precisely, it is assumed that the density (i) is smooth
in a neighborhood of and (ii) has -exponent at level
. Condition (i) ensures that the density can be estimated at a
standard nonparametric rate and condition (ii) is similar to Tsybakov's margin
assumption which is stated for the classification framework. Under these
assumptions, we derive optimal rates of convergence for plug-in estimators.
Explicit convergence rates are given for plug-in estimators based on kernel
density estimators when the underlying measure is the Lebesgue measure. Lower
bounds proving optimality of the rates in a minimax sense when the density is
H\"older smooth are also provided.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ184 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Minimax and Adaptive Inference in Nonparametric Function Estimation
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in
both parametric and nonparametric inference. This article discusses minimaxity
and adaptive minimaxity in nonparametric function estimation. Three
interrelated problems, function estimation under global integrated squared
error, estimation under pointwise squared error, and nonparametric confidence
intervals, are considered. Shrinkage is pivotal in the development of both the
minimax theory and the adaptation theory. While the three problems are closely
connected and the minimax theories bear some similarities, the adaptation
theories are strikingly different. For example, in a sharp contrast to adaptive
point estimation, in many common settings there do not exist nonparametric
confidence intervals that adapt to the unknown smoothness of the underlying
function. A concise account of these theories is given. The connections as well
as differences among these problems are discussed and illustrated through
examples.Comment: Published in at http://dx.doi.org/10.1214/11-STS355 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Adaptation to lowest density regions with application to support recovery
A scheme for locally adaptive bandwidth selection is proposed which
sensitively shrinks the bandwidth of a kernel estimator at lowest density
regions such as the support boundary which are unknown to the statistician. In
case of a H\"{o}lder continuous density, this locally minimax-optimal bandwidth
is shown to be smaller than the usual rate, even in case of homogeneous
smoothness. Some new type of risk bound with respect to a density-dependent
standardized loss of this estimator is established. This bound is fully
nonasymptotic and allows to deduce convergence rates at lowest density regions
that can be substantially faster than . It is complemented by a
weighted minimax lower bound which splits into two regimes depending on the
value of the density. The new estimator adapts into the second regime, and it
is shown that simultaneous adaptation into the fastest regime is not possible
in principle as long as the H\"{o}lder exponent is unknown. Consequences on
plug-in rules for support recovery are worked out in detail. In contrast to
those with classical density estimators, the plug-in rules based on the new
construction are minimax-optimal, up to some logarithmic factor.Comment: Published at http://dx.doi.org/10.1214/15-AOS1366 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models
This note uses a simple example to show how moment inequality models used in
the empirical economics literature lead to general minimax relative efficiency
comparisons. The main point is that such models involve inference on a low
dimensional parameter, which leads naturally to a definition of "distance"
that, in full generality, would be arbitrary in minimax testing problems. This
definition of distance is justified by the fact that it leads to a duality
between minimaxity of confidence intervals and tests, which does not hold for
other definitions of distance. Thus, the use of moment inequalities for
inference in a low dimensional parametric model places additional structure on
the testing problem, which leads to stronger conclusions regarding minimax
relative efficiency than would otherwise be possible
Nonparametric estimation over shrinking neighborhoods: Superefficiency and adaptation
A theory of superefficiency and adaptation is developed under flexible
performance measures which give a multiresolution view of risk and bridge the
gap between pointwise and global estimation. This theory provides a useful
benchmark for the evaluation of spatially adaptive estimators and shows that
the possible degree of superefficiency for minimax rate optimal estimators
critically depends on the size of the neighborhood over which the risk is
measured. Wavelet procedures are given which adapt rate optimally for given
shrinking neighborhoods including the extreme cases of mean squared error at a
point and mean integrated squared error over the whole interval. These adaptive
procedures are based on a new wavelet block thresholding scheme which combines
both the commonly used horizontal blocking of wavelet coefficients (at the same
resolution level) and vertical blocking of coefficients (across different
resolution levels).Comment: Published at http://dx.doi.org/10.1214/009053604000000832 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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