322 research outputs found

### Estimation of sums of random variables: Examples and information bounds

This paper concerns the estimation of sums of functions of observable and
unobservable variables. Lower bounds for the asymptotic variance and a
convolution theorem are derived in general finite- and infinite-dimensional
models. An explicit relationship is established between efficient influence
functions for the estimation of sums of variables and the estimation of their
means. Certain ``plug-in'' estimators are proved to be asymptotically efficient
in finite-dimensional models, while ``$u,v$'' estimators of Robbins are proved
to be efficient in infinite-dimensional mixture models. Examples include
certain species, network and data confidentiality problems.Comment: Published at http://dx.doi.org/10.1214/009053605000000390 in the
Annals of Statistics (http://www.imstat.org/aos/) 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

### Discussion: One-step sparse estimates in nonconcave penalized likelihood models

Discussion of ``One-step sparse estimates in nonconcave penalized likelihood
models'' [arXiv:0808.1012]Comment: Published in at http://dx.doi.org/10.1214/07-AOS0316C the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### A General Framework of Dual Certificate Analysis for Structured Sparse Recovery Problems

This paper develops a general theoretical framework to analyze structured
sparse recovery problems using the notation of dual certificate. Although
certain aspects of the dual certificate idea have already been used in some
previous work, due to the lack of a general and coherent theory, the analysis
has so far only been carried out in limited scopes for specific problems. In
this context the current paper makes two contributions. First, we introduce a
general definition of dual certificate, which we then use to develop a unified
theory of sparse recovery analysis for convex programming. Second, we present a
class of structured sparsity regularization called structured Lasso for which
calculations can be readily performed under our theoretical framework. This new
theory includes many seemingly loosely related previous work as special cases;
it also implies new results that improve existing ones even for standard
formulations such as L1 regularization

### General maximum likelihood empirical Bayes estimation of normal means

We propose a general maximum likelihood empirical Bayes (GMLEB) method for
the estimation of a mean vector based on observations with i.i.d. normal
errors. We prove that under mild moment conditions on the unknown means, the
average mean squared error (MSE) of the GMLEB is within an infinitesimal
fraction of the minimum average MSE among all separable estimators which use a
single deterministic estimating function on individual observations, provided
that the risk is of greater order than $(\log n)^5/n$. We also prove that the
GMLEB is uniformly approximately minimax in regular and weak $\ell_p$ balls
when the order of the length-normalized norm of the unknown means is between
$(\log n)^{\kappa_1}/n^{1/(p\wedge2)}$ and $n/(\log n)^{\kappa_2}$. Simulation
experiments demonstrate that the GMLEB outperforms the James--Stein and several
state-of-the-art threshold estimators in a wide range of settings without much
down side.Comment: Published in at http://dx.doi.org/10.1214/08-AOS638 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Calibrated Elastic Regularization in Matrix Completion

This paper concerns the problem of matrix completion, which is to estimate a
matrix from observations in a small subset of indices. We propose a calibrated
spectrum elastic net method with a sum of the nuclear and Frobenius penalties
and develop an iterative algorithm to solve the convex minimization problem.
The iterative algorithm alternates between imputing the missing entries in the
incomplete matrix by the current guess and estimating the matrix by a scaled
soft-thresholding singular value decomposition of the imputed matrix until the
resulting matrix converges. A calibration step follows to correct the bias
caused by the Frobenius penalty. Under proper coherence conditions and for
suitable penalties levels, we prove that the proposed estimator achieves an
error bound of nearly optimal order and in proportion to the noise level. This
provides a unified analysis of the noisy and noiseless matrix completion
problems. Simulation results are presented to compare our proposal with
previous ones.Comment: 9 pages; Advances in Neural Information Processing Systems, NIPS 201

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