13,833 research outputs found
Adaptive Nonparametric Empirical Bayes Estimation Via Wavelet Series: the Minimax Study
In the present paper, we derive lower bounds for the risk of the
nonparametric empirical Bayes estimators. In order to attain the optimal
convergence rate, we propose generalization of the linear empirical Bayes
estimation method which takes advantage of the flexibility of the wavelet
techniques. We present an empirical Bayes estimator as a wavelet series
expansion and estimate coefficients by minimizing the prior risk of the
estimator. As a result, estimation of wavelet coefficients requires solution of
a well-posed low-dimensional sparse system of linear equations. The dimension
of the system depends on the size of wavelet support and smoothness of the
Bayes estimator. An adaptive choice of the resolution level is carried out
using Lepski (1997) method. The method is computationally efficient and
provides asymptotically optimal adaptive EB estimators. The theory is
supplemented by numerous examples
A nonparametric empirical Bayes approach to covariance matrix estimation
We propose an empirical Bayes method to estimate high-dimensional covariance
matrices. Our procedure centers on vectorizing the covariance matrix and
treating matrix estimation as a vector estimation problem. Drawing from the
compound decision theory literature, we introduce a new class of decision rules
that generalizes several existing procedures. We then use a nonparametric
empirical Bayes g-modeling approach to estimate the oracle optimal rule in that
class. This allows us to let the data itself determine how best to shrink the
estimator, rather than shrinking in a pre-determined direction such as toward a
diagonal matrix. Simulation results and a gene expression network analysis
shows that our approach can outperform a number of state-of-the-art proposals
in a wide range of settings, sometimes substantially.Comment: 20 pages, 4 figure
Bandwidth choice for nonparametric classification
It is shown that, for kernel-based classification with univariate
distributions and two populations, optimal bandwidth choice has a dichotomous
character. If the two densities cross at just one point, where their curvatures
have the same signs, then minimum Bayes risk is achieved using bandwidths which
are an order of magnitude larger than those which minimize pointwise estimation
error. On the other hand, if the curvature signs are different, or if there are
multiple crossing points, then bandwidths of conventional size are generally
appropriate. The range of different modes of behavior is narrower in
multivariate settings. There, the optimal size of bandwidth is generally the
same as that which is appropriate for pointwise density estimation. These
properties motivate empirical rules for bandwidth choice.Comment: Published at http://dx.doi.org/10.1214/009053604000000959 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
General empirical Bayes wavelet methods and exactly adaptive minimax estimation
In many statistical problems, stochastic signals can be represented as a
sequence of noisy wavelet coefficients. In this paper, we develop general
empirical Bayes methods for the estimation of true signal. Our estimators
approximate certain oracle separable rules and achieve adaptation to ideal
risks and exact minimax risks in broad collections of classes of signals. In
particular, our estimators are uniformly adaptive to the minimum risk of
separable estimators and the exact minimax risks simultaneously in Besov balls
of all smoothness and shape indices, and they are uniformly superefficient in
convergence rates in all compact sets in Besov spaces with a finite secondary
shape parameter. Furthermore, in classes nested between Besov balls of the same
smoothness index, our estimators dominate threshold and James-Stein estimators
within an infinitesimal fraction of the minimax risks. More general block
empirical Bayes estimators are developed. Both white noise with drift and
nonparametric regression are considered.Comment: Published at http://dx.doi.org/10.1214/009053604000000995 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bandwidth choice for nonparametric classification
It is shown that, for kernel-based classification with univariate
distributions and two populations, optimal bandwidth choice has a dichotomous
character. If the two densities cross at just one point, where their curvatures
have the same signs, then minimum Bayes risk is achieved using bandwidths which
are an order of magnitude larger than those which minimize pointwise estimation
error. On the other hand, if the curvature signs are different, or if there are
multiple crossing points, then bandwidths of conventional size are generally
appropriate. The range of different modes of behavior is narrower in
multivariate settings. There, the optimal size of bandwidth is generally the
same as that which is appropriate for pointwise density estimation. These
properties motivate empirical rules for bandwidth choice
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