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## Shrinkage Estimators for Covariance Matrices

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em /12/2001
Português

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Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecifics tion of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases...

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## Functional Connectivity: Shrinkage Estimation and Randomization Test

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Português

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We develop new statistical methods for estimating functional connectivity between components of a multivariate time series and for testing differences in functional connectivity across experimental conditions. Here, we characterize functional connectivity by partial coherence, which identifies the frequency band (or bands) that drives the direct linear association between any pair of components of a multivariate time series after removing the linear effects of the other components. Partial coherence can be efficiently estimated using the inverse of the spectral density matrix. However, when the number of components is large and the components of the multivariate time series are highly correlated, the spectral density matrix estimate may be numerically unstable and consequently gives partial coherence estimates that are highly variable. To address the problem of numerical instability, we propose a shrinkage-based estimator which is a weighted average of a smoothed periodogram estimator and a scaled identity matrix with frequency-specific weight computed objectively so that the resulting shrinkage estimator minimizes the mean-squared error criterion. Compared to typical smoothing-based estimators, the shrinkage estimator is more computationally stable and gives a lower mean squared error. In addition...

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## Properties of preliminary test estimators and shrinkage estimators for evaluating multiple exposures – Application to questionnaire data from the SONIC study

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em 01/08/2011
Português

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Epidemiology studies increasingly examine multiple exposures in relation to disease by selecting the exposures of interest in a thematic manner. For example, sun exposure, sunburn, and sun protection behavior could be themes for an investigation of sun-related exposures. Several studies now use pre-defined linear combinations of the exposures pertaining to the themes to estimate the effects of the individual exposures. Such analyses may improve the precision of the exposure effects, but they can lead to inflated bias and type I errors when the linear combinations are inaccurate. We investigate preliminary test estimators and empirical Bayes type shrinkage estimators as alternative approaches when it is desirable to exploit the thematic choice of exposures, but the accuracy of the pre-defined linear combinations is unknown. We show that the two types of estimator are intimately related under certain assumptions. The shrinkage estimator derived under the assumption of an exchangeable prior distribution gives precise estimates and is robust to misspecifications of the user-defined linear combinations. The precision gains and robustness of the shrinkage estimation approach are illustrated using data from the SONIC study, where the exposures are the individual questionnaire items and the outcome is (log) total back nevus count.

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## The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em //2011
Português

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We propose a new penalized method for variable selection and estimation that explicitly incorporates the correlation patterns among predictors. This method is based on a combination of the minimax concave penalty and Laplacian quadratic associated with a graph as the penalty function. We call it the sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave penalty for encouraging sparsity and Laplacian quadratic penalty for promoting smoothness among coefficients associated with the correlated predictors. The SLS has a generalized grouping property with respect to the graph represented by the Laplacian quadratic. We show that the SLS possesses an oracle property in the sense that it is selection consistent and equal to the oracle Laplacian shrinkage estimator with high probability. This result holds in sparse, high-dimensional settings with p ≫ n under reasonable conditions. We derive a coordinate descent algorithm for computing the SLS estimates. Simulation studies are conducted to evaluate the performance of the SLS method and a real data example is used to illustrate its application.

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## A Sparse Structured Shrinkage Estimator for Nonparametric Varying-Coefficient Model with an Application in Genomics

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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Many problems in genomics are related to variable selection where high-dimensional genomic data are treated as covariates. Such genomic covariates often have certain structures and can be represented as vertices of an undirected graph. Biological processes also vary as functions depending upon some biological state, such as time. High-dimensional variable selection where covariates are graph-structured and underlying model is nonparametric presents an important but largely unaddressed statistical challenge. Motivated by the problem of regression-based motif discovery, we consider the problem of variable selection for high-dimensional nonparametric varying-coefficient models and introduce a sparse structured shrinkage (SSS) estimator based on basis function expansions and a novel smoothed penalty function. We present an efficient algorithm for computing the SSS estimator. Results on model selection consistency and estimation bounds are derived. Moreover, finite-sample performances are studied via simulations, and the effects of high-dimensionality and structural information of the covariates are especially highlighted. We apply our method to motif finding problem using a yeast cell-cycle gene expression dataset and word counts in genes’ promoter sequences. Our results demonstrate that the proposed method can result in better variable selection and prediction for high-dimensional regression when the underlying model is nonparametric and covariates are structured. Supplemental materials for the article are available online.

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## SHAVE: shrinkage estimator measured for multiple visits increases power in GWAS of quantitative traits

Fonte: Nature Publishing Group
Publicador: Nature Publishing Group

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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Measurement error and biological variability generate distortions in quantitative phenotypic data. In longitudinal studies with repeated measurements, the multiple measurements provide a route to reduce noise and correspondingly increase the strength of signals in genome-wide association studies (GWAS).To optimize noise correction, we have developed Shrunken Average (SHAVE), an approach using a Bayesian Shrinkage estimator. This estimator uses regression toward the mean for every individual as a function of (1) their average across visits; (2) their number of visits; and (3) the correlation between visits. Computer simulations support an increase in power, with results very similar to those expected by the assumptions of the model. The method was applied to a real data set for 14 anthropomorphic traits in ∼6000 individuals enrolled in the SardiNIA project, with up to three visits (measurements) for each participant. Results show that additional measurements have a large impact on the strength of GWAS signals, especially when participants have different number of visits, with SHAVE showing a clear increase in power relative to single visits. In addition, we have derived a relation to assess the improvement in power as a function of number of visits and correlation between visits. It can also be applied in the optimization of experimental designs or usage of measuring devices. SHAVE is fast and easy to run...

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## Testing Departure from Additivity in Tukey’s Model using Shrinkage: Application to a Longitudinal Setting

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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While there has been extensive research developing gene-environment interaction (GEI) methods in case-control studies, little attention has been given to sparse and efficient modeling of GEI in longitudinal studies. In a two-way table for GEI with rows and columns as categorical variables, a conventional saturated interaction model involves estimation of a specific parameter for each cell, with constraints ensuring identifiability. The estimates are unbiased but are potentially inefficient because the number of parameters to be estimated can grow quickly with increasing categories of row/column factors. On the other hand, Tukey’s one degree of freedom (df) model for non-additivity treats the interaction term as a scaled product of row and column main effects. Due to the parsimonious form of interaction, the interaction estimate leads to enhanced efficiency and the corresponding test could lead to increased power. Unfortunately, Tukey’s model gives biased estimates and low power if the model is misspecified. When screening multiple GEIs where each genetic and environmental marker may exhibit a distinct interaction pattern, a robust estimator for interaction is important for GEI detection. We propose a shrinkage estimator for interaction effects that combines estimates from both Tukey’s and saturated interaction models and use the corresponding Wald test for testing interaction in a longitudinal setting. The proposed estimator is robust to misspecification of interaction structure. We illustrate the proposed methods using two longitudinal studies — the Normative Aging Study and the Multi-Ethnic Study of Atherosclerosis.

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## A well conditioned estimator for large dimensional covariance matrices

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: Trabalho em Andamento
Formato: application/octet-stream; application/octet-stream; application/pdf

Publicado em /11/2000
Português

Relevância na Pesquisa

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#Condition number#Covariance matrix estimation#Empirical Bayes#General asymptotics#Shrinkage#Estadística

Many economic problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For largedimensional covariance matrices, the usual estimator -the sample covariance matrix- is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte-Carlo confirm that the asymptotic results tend to hold well in finite sample.

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## Efficient estimation using the characteristic function : theory and applications with high frequency data

Fonte: Université de Montréal
Publicador: Université de Montréal

Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation

Português

Relevância na Pesquisa

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#Integrated volatility#Volatilité intégré#Method of moment#Méthode des moments#Microstructure noise#Bruit de microstructure#Realized Kernel#Volatilité réalisée à Noyaux#Shrinkage estimator#Combinaison linéaire optimale d'estimateur#Continuum of moment conditions

Nous abordons deux sujets distincts dans cette thèse: l'estimation de la volatilité des prix d'actifs financiers à partir des données à haute fréquence, et l'estimation des paramétres d'un processus aléatoire à partir de sa fonction caractéristique.
Le chapitre 1 s'intéresse à l'estimation de la volatilité des prix d'actifs. Nous supposons que les données à haute fréquence disponibles sont entachées de bruit de microstructure. Les propriétés que l'on prête au bruit sont déterminantes dans le choix de l'estimateur de la volatilité. Dans ce chapitre, nous spécifions un nouveau modèle dynamique pour le bruit de microstructure qui intègre trois propriétés importantes: (i) le bruit peut être autocorrélé, (ii) le retard maximal au delà duquel l'autocorrélation est nulle peut être une fonction croissante de la fréquence journalière d'observations; (iii) le bruit peut avoir une composante correlée avec le rendement efficient. Cette dernière composante est alors dite endogène. Ce modèle se différencie de ceux existant en ceci qu'il implique que l'autocorrélation d'ordre 1 du bruit converge vers 1 lorsque la fréquence journalière d'observation tend vers l'infini.
Nous utilisons le cadre semi-paramétrique ainsi défini pour dériver un nouvel estimateur de la volatilité intégrée baptisée "estimateur shrinkage". Cet estimateur se présente sous la forme d'une combinaison linéaire optimale de deux estimateurs aux propriétés différentes...

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## Shrinkage Algorithms for MMSE Covariance Estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/07/2009
Português

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We address covariance estimation in the sense of minimum mean-squared error
(MMSE) for Gaussian samples. Specifically, we consider shrinkage methods which
are suitable for high dimensional problems with a small number of samples
(large p small n). First, we improve on the Ledoit-Wolf (LW) method by
conditioning on a sufficient statistic. By the Rao-Blackwell theorem, this
yields a new estimator called RBLW, whose mean-squared error dominates that of
LW for Gaussian variables. Second, to further reduce the estimation error, we
propose an iterative approach which approximates the clairvoyant shrinkage
estimator. Convergence of this iterative method is established and a closed
form expression for the limit is determined, which is referred to as the oracle
approximating shrinkage (OAS) estimator. Both RBLW and OAS estimators have
simple expressions and are easily implemented. Although the two methods are
developed from different persepctives, their structure is identical up to
specified constants. The RBLW estimator provably dominates the LW method.
Numerical simulations demonstrate that the OAS approach can perform even better
than RBLW, especially when n is much less than p. We also demonstrate the
performance of these techniques in the context of adaptive beamforming.

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## Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

48.67707%

#Mathematics - Statistics Theory#Mathematics - Probability#Quantitative Finance - Statistical Finance

In this work we construct an optimal shrinkage estimator for the precision
matrix in high dimensions. We consider the general asymptotics when the number
of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so
that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated
directly, without inverting the corresponding estimator for the covariance
matrix. The recent results from the random matrix theory allow us to find the
asymptotic deterministic equivalents of the optimal shrinkage intensities and
estimate them consistently. The resulting distribution-free estimator has
almost surely the minimum Frobenius loss. Additionally, we prove that the
Frobenius norms of the inverse and of the pseudo-inverse sample covariance
matrices tend almost surely to deterministic quantities and estimate them
consistently. At the end, a simulation is provided where the suggested
estimator is compared with the estimators for the precision matrix proposed in
the literature. The optimal shrinkage estimator shows significant improvement
and robustness even for non-normally distributed data.; Comment: 26 pages, 5 figures. This version includes the case c>1 with the
generalized inverse of the sample covariance matrix. The abstract was updated
accordingly

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## The sparse Laplacian shrinkage estimator for high-dimensional regression

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2011
Português

Relevância na Pesquisa

48.091084%

We propose a new penalized method for variable selection and estimation that
explicitly incorporates the correlation patterns among predictors. This method
is based on a combination of the minimax concave penalty and Laplacian
quadratic associated with a graph as the penalty function. We call it the
sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave
penalty for encouraging sparsity and Laplacian quadratic penalty for promoting
smoothness among coefficients associated with the correlated predictors. The
SLS has a generalized grouping property with respect to the graph represented
by the Laplacian quadratic. We show that the SLS possesses an oracle property
in the sense that it is selection consistent and equal to the oracle Laplacian
shrinkage estimator with high probability. This result holds in sparse,
high-dimensional settings with p >> n under reasonable conditions. We derive a
coordinate descent algorithm for computing the SLS estimates. Simulation
studies are conducted to evaluate the performance of the SLS method and a real
data example is used to illustrate its application.; Comment: Published in at http://dx.doi.org/10.1214/11-AOS897 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## The generalized shrinkage estimator for the analysis of functional connectivity of brain signals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2011
Português

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We develop a new statistical method for estimating functional connectivity
between neurophysiological signals represented by a multivariate time series.
We use partial coherence as the measure of functional connectivity. Partial
coherence identifies the frequency bands that drive the direct linear
association between any pair of channels. To estimate partial coherence, one
would first need an estimate of the spectral density matrix of the multivariate
time series. Parametric estimators of the spectral density matrix provide good
frequency resolution but could be sensitive when the parametric model is
misspecified. Smoothing-based nonparametric estimators are robust to model
misspecification and are consistent but may have poor frequency resolution. In
this work, we develop the generalized shrinkage estimator, which is a weighted
average of a parametric estimator and a nonparametric estimator. The optimal
weights are frequency-specific and derived under the quadratic risk criterion
so that the estimator, either the parametric estimator or the nonparametric
estimator, that performs better at a particular frequency receives heavier
weight. We validate the proposed estimator in a simulation study and apply it
on electroencephalogram recordings from a visual-motor experiment.; Comment: Published in at http://dx.doi.org/10.1214/10-AOAS396 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Non-parametric shrinkage mean estimation for quadratic loss functions with unknown covariance matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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In this paper, a shrinkage estimator for the population mean is proposed
under known quadratic loss functions with unknown covariance matrices. The new
estimator is non-parametric in the sense that it does not assume a specific
parametric distribution for the data and it does not require the prior
information on the population covariance matrix. Analytical results on the
improvement of the proposed shrinkage estimator are provided and some
corresponding asymptotic properties are also derived. Finally, we demonstrate
the practical improvement of the proposed method over existing methods through
extensive simulation studies and real data analysis. Keywords: High-dimensional
data; Shrinkage estimator; Large $p$ small $n$; $U$-statistic.; Comment: Some technical parts of Theorem 3.1 and 3.2 were corrected in this
version

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## Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

38.068452%

We present a procedure for effective estimation of entropy and mutual
information from small-sample data, and apply it to the problem of inferring
high-dimensional gene association networks. Specifically, we develop a
James-Stein-type shrinkage estimator, resulting in a procedure that is highly
efficient statistically as well as computationally. Despite its simplicity, we
show that it outperforms eight other entropy estimation procedures across a
diverse range of sampling scenarios and data-generating models, even in cases
of severe undersampling. We illustrate the approach by analyzing E. coli gene
expression data and computing an entropy-based gene-association network from
gene expression data. A computer program is available that implements the
proposed shrinkage estimator.; Comment: 18 pages, 3 figures, 1 table

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## A shrinkage estimation for large dimensional precision matrices using random matrix theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

38.4184%

In this paper, a new ridge-type shrinkage estimator for the precision matrix
has been proposed. The asymptotic optimal shrinkage coefficients and the
theoretical loss were derived. Data-driven estimators for the shrinkage
coefficients were also conducted based on the asymptotic results deriving from
random matrix theories. The new estimator which has a simple explicit formula
is distribution-free and applicable to situation where the dimension of
observation is greater than the sample size. Further, no assumptions are
required on the structure of the population covariance matrix or the precision
matrix. Finally, numerical studies are conducted to examine the performances of
the new estimator and existing methods for a wide range of settings.; Comment: This paper has been withdrawn by the author due to substantial
contents will be updated

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## On the Strong Convergence of the Optimal Linear Shrinkage Estimator for Large Dimensional Covariance Matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

48.2244%

#Mathematics - Statistics Theory#Mathematics - Probability#Quantitative Finance - Statistical Finance#60B20, 62H12, 62G20, 62G30

In this work we construct an optimal linear shrinkage estimator for the
covariance matrix in high dimensions. The recent results from the random matrix
theory allow us to find the asymptotic deterministic equivalents of the optimal
shrinkage intensities and estimate them consistently. The developed
distribution-free estimators obey almost surely the smallest Frobenius loss
over all linear shrinkage estimators for the covariance matrix. The case we
consider includes the number of variables $p\rightarrow\infty$ and the sample
size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$.
Additionally, we prove that the Frobenius norm of the sample covariance matrix
tends almost surely to a deterministic quantity which can be consistently
estimated.; Comment: 21 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1308.0931, revised version (Journal of Multivariate Analysis)

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## Generalized robust shrinkage estimator and its application to STAP detection problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

48.77419%

Recently, in the context of covariance matrix estimation, in order to improve
as well as to regularize the performance of the Tyler's estimator [1] also
called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator
has been introduced in [3]. First, this work extends the results of [3,4] by
giving the general solution of the "shrinkage" fixed-point algorithm. Secondly,
by analyzing this solution, called the generalized robust shrinkage estimator,
we prove that this solution converges to a unique solution when the shrinkage
parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE
with the trace of its inverse equal to the dimension of the problem. This
general result allows one to give another interpretation of the FPE and more
generally, on the Maximum Likelihood approach for covariance matrix estimation
when constraints are added. Then, some simulations illustrate our theoretical
results as well as the way to choose an optimal shrinkage factor. Finally, this
work is applied to a Space-Time Adaptive Processing (STAP) detection problem on
real STAP data.

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## Adaptive Monotone Shrinkage for Regression

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/05/2015
Português

Relevância na Pesquisa

38.673088%

We develop an adaptive monotone shrinkage estimator for regression models
with the following characteristics: i) dense coefficients with small but
important effects; ii) a priori ordering that indicates the probable predictive
importance of the features. We capture both properties with an empirical Bayes
estimator that shrinks coefficients monotonically with respect to their
anticipated importance. This estimator can be rapidly computed using a version
of Pool-Adjacent-Violators algorithm. We show that the proposed monotone
shrinkage approach is competitive with the class of all Bayesian estimators
that share the prior information. We further observe that the estimator also
minimizes Stein's unbiased risk estimate. Along with our key result that the
estimator mimics the oracle Bayes rule under an order assumption, we also prove
that the estimator is robust. Even without the order assumption, our estimator
mimics the best performance of a large family of estimators that includes the
least squares estimator, constant-$\lambda$ ridge estimator, James-Stein
estimator, etc. All the theoretical results are non-asymptotic. Simulation
results and data analysis from a model for text processing are provided to
support the theory.; Comment: Appearing in Uncertainty in Artificial Intelligence (UAI) 2014

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## Shrinkage Estimation of the Power Spectrum Covariance Matrix

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

38.314707%

We seek to improve estimates of the power spectrum covariance matrix from a
limited number of simulations by employing a novel statistical technique known
as shrinkage estimation. The shrinkage technique optimally combines an
empirical estimate of the covariance with a model (the target) to minimize the
total mean squared error compared to the true underlying covariance. We test
this technique on N-body simulations and evaluate its performance by estimating
cosmological parameters. Using a simple diagonal target, we show that the
shrinkage estimator significantly outperforms both the empirical covariance and
the target individually when using a small number of simulations. We find that
reducing noise in the covariance estimate is essential for properly estimating
the values of cosmological parameters as well as their confidence intervals. We
extend our method to the jackknife covariance estimator and again find
significant improvement, though simulations give better results. Even for
thousands of simulations we still find evidence that our method improves
estimation of the covariance matrix. Because our method is simple, requires
negligible additional numerical effort, and produces superior results, we
always advocate shrinkage estimation for the covariance of the power spectrum
and other large-scale structure measurements when purely theoretical modeling
of the covariance is insufficient.; Comment: 9 pages...

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