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Shrinkage Estimators for Covariance Matrices

Daniels, Michael J.; Kass, Robert E.
Tipo: Artigo de Revista Científica
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38.981016%
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...

Functional Connectivity: Shrinkage Estimation and Randomization Test

Fiecas, Mark; Ombao, Hernando; Linkletter, Crystal; Thompson, Wesley; Sanes, Jerome
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.537786%
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...

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

Satagopan, Jaya M.; Zhou, Qin; Oliveria, Susan A.; Dusza, Stephen W.; Weinstock, Martin A.; Berwick, Marianne; Halpern, Allan C.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
38.2244%
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.

The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

Huang, Jian; Ma, Shuangge; Li, Hongzhe; Zhang, Cun-Hui
Tipo: Artigo de Revista Científica
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.

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

Daye, Z. John; Xie, Jichun; Li, Hongzhe
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.068452%
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.

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

Meirelles, Osorio D; Ding, Jun; Tanaka, Toshiko; Sanna, Serena; Yang, Hsih-Te; Dudekula, Dawood B; Cucca, Francesco; Ferrucci, Luigi; Abecasis, Goncalo; Schlessinger, David
Fonte: Nature Publishing Group Publicador: Nature Publishing Group
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
48.06845%
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...

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

Ko, Yi-An; Mukherjee, Bhramar; Smith, Jennifer A.; Park, Sung Kyun; Kardia, Sharon L.R.; Allison, Matthew A.; Vokonas, Pantel S.; Chen, Jinbo; Diez-Roux, Ana V.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.068452%
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.

A well conditioned estimator for large dimensional covariance matrices

Ledoit, Olivier; Wolf, Michael
Tipo: Trabalho em Andamento Formato: application/octet-stream; application/octet-stream; application/pdf
Relevância na Pesquisa
37.855137%
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.

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

Kotchoni, Rachidi
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation
Português
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48.945894%
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...

Shrinkage Algorithms for MMSE Covariance Estimation

Chen, Yilun; Wiesel, Ami; Eldar, Yonina C.; Hero III, Alfred O.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
38.537786%
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.

Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix

Bodnar, Taras; Gupta, Arjun K.; Parolya, Nestor
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
48.67707%
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

The sparse Laplacian shrinkage estimator for high-dimensional regression

Huang, Jian; Ma, Shuangge; Li, Hongzhe; Zhang, Cun-Hui
Tipo: Artigo de Revista Científica
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)

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

Fiecas, Mark; Ombao, Hernando
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.405845%
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)

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

Wang, Cheng; Tong, Tiejun; Cao, Longbing; Miao, Baiqi
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.537786%
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

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

Hausser, Jean; Strimmer, Korbinian
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

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

Wang, Cheng; Pan, Guangming; Cao, Longbing
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

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

Bodnar, Taras; Gupta, Arjun K.; Parolya, Nestor
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
48.2244%
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)

Generalized robust shrinkage estimator and its application to STAP detection problem

Pascal, Frederic; Chitour, Yacine; Quek, Yihui
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.

Ma, Zhuang; Foster, Dean; Stine, Robert
Tipo: Artigo de Revista Científica
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