5,910 research outputs found
Comment: Microarrays, Empirical Bayes and the Two-Group Model
Comment on ``Microarrays, Empirical Bayes and the Two-Group Model''
[arXiv:0808.0572]Comment: Published in at http://dx.doi.org/10.1214/07-STS236C the Statistical
Science (http://www.imstat.org/sts/) 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
Comment: Microarrays, Empirical Bayes and the Two-Groups Model
Comment on ``Microarrays, Empirical Bayes and the Two-Groups Model''
[arXiv:0808.0572]Comment: Published in at http://dx.doi.org/10.1214/07-STS236A the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery
This paper establishes a sharp condition on the restricted isometry property
(RIP) for both the sparse signal recovery and low-rank matrix recovery. It is
shown that if the measurement matrix satisfies the RIP condition
, then all -sparse signals can be recovered exactly
via the constrained minimization based on . Similarly, if
the linear map satisfies the RIP condition ,
then all matrices of rank at most can be recovered exactly via the
constrained nuclear norm minimization based on . Furthermore, in
both cases it is not possible to do so in general when the condition does not
hold. In addition, noisy cases are considered and oracle inequalities are given
under the sharp RIP condition.Comment: to appear in Applied and Computational Harmonic Analysis (2012
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