46,225 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
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
Structured Matrix Completion with Applications to Genomic Data Integration
Matrix completion has attracted significant recent attention in many fields
including statistics, applied mathematics and electrical engineering. Current
literature on matrix completion focuses primarily on independent sampling
models under which the individual observed entries are sampled independently.
Motivated by applications in genomic data integration, we propose a new
framework of structured matrix completion (SMC) to treat structured missingness
by design. Specifically, our proposed method aims at efficient matrix recovery
when a subset of the rows and columns of an approximately low-rank matrix are
observed. We provide theoretical justification for the proposed SMC method and
derive lower bound for the estimation errors, which together establish the
optimal rate of recovery over certain classes of approximately low-rank
matrices. Simulation studies show that the method performs well in finite
sample under a variety of configurations. The method is applied to integrate
several ovarian cancer genomic studies with different extent of genomic
measurements, which enables us to construct more accurate prediction rules for
ovarian cancer survival.Comment: Accepted for publication in Journal of the American Statistical
Associatio
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
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
Adaptive variance function estimation in heteroscedastic nonparametric regression
We consider a wavelet thresholding approach to adaptive variance function
estimation in heteroscedastic nonparametric regression. A data-driven estimator
is constructed by applying wavelet thresholding to the squared first-order
differences of the observations. We show that the variance function estimator
is nearly optimally adaptive to the smoothness of both the mean and variance
functions. The estimator is shown to achieve the optimal adaptive rate of
convergence under the pointwise squared error simultaneously over a range of
smoothness classes. The estimator is also adaptively within a logarithmic
factor of the minimax risk under the global mean integrated squared error over
a collection of spatially inhomogeneous function classes. Numerical
implementation and simulation results are also discussed.Comment: Published in at http://dx.doi.org/10.1214/07-AOS509 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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