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Author : Debra Lynn Hydorn
Publisher :
ISBN 13 :
Total Pages : 364 pages
Book Rating : 4.3/5 (91 download)
Download or read book On Estimating the Eigenvalues of a Covariance Matrix written by Debra Lynn Hydorn and published by . This book was released on 1993 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Aygul Zagidullina
Publisher : Springer Nature
ISBN 13 : 3030800652
Total Pages : 123 pages
Book Rating : 4.0/5 (38 download)
Download or read book High-Dimensional Covariance Matrix Estimation written by Aygul Zagidullina and published by Springer Nature. This book was released on 2021-10-29 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents covariance matrix estimation and related aspects of random matrix theory. It focuses on the sample covariance matrix estimator and provides a holistic description of its properties under two asymptotic regimes: the traditional one, and the high-dimensional regime that better fits the big data context. It draws attention to the deficiencies of standard statistical tools when used in the high-dimensional setting, and introduces the basic concepts and major results related to spectral statistics and random matrix theory under high-dimensional asymptotics in an understandable and reader-friendly way. The aim of this book is to inspire applied statisticians, econometricians, and machine learning practitioners who analyze high-dimensional data to apply the recent developments in their work.
Author : Mohsen Pourahmadi
Publisher : John Wiley & Sons
ISBN 13 : 1118034295
Total Pages : 204 pages
Book Rating : 4.1/5 (18 download)
Download or read book High-Dimensional Covariance Estimation written by Mohsen Pourahmadi and published by John Wiley & Sons. This book was released on 2013-06-24 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Methods for estimating sparse and large covariance matrices Covariance and correlation matrices play fundamental roles in every aspect of the analysis of multivariate data collected from a variety of fields including business and economics, health care, engineering, and environmental and physical sciences. High-Dimensional Covariance Estimation provides accessible and comprehensive coverage of the classical and modern approaches for estimating covariance matrices as well as their applications to the rapidly developing areas lying at the intersection of statistics and machine learning. Recently, the classical sample covariance methodologies have been modified and improved upon to meet the needs of statisticians and researchers dealing with large correlated datasets. High-Dimensional Covariance Estimation focuses on the methodologies based on shrinkage, thresholding, and penalized likelihood with applications to Gaussian graphical models, prediction, and mean-variance portfolio management. The book relies heavily on regression-based ideas and interpretations to connect and unify many existing methods and algorithms for the task. High-Dimensional Covariance Estimation features chapters on: Data, Sparsity, and Regularization Regularizing the Eigenstructure Banding, Tapering, and Thresholding Covariance Matrices Sparse Gaussian Graphical Models Multivariate Regression The book is an ideal resource for researchers in statistics, mathematics, business and economics, computer sciences, and engineering, as well as a useful text or supplement for graduate-level courses in multivariate analysis, covariance estimation, statistical learning, and high-dimensional data analysis.
Author : John Edward Vetter
Publisher :
ISBN 13 :
Total Pages : 294 pages
Book Rating : 4.:/5 (373 download)
Download or read book Estimation of Covariance Matrices Using Influence of Eigenvectors and Eigenvalues written by John Edward Vetter and published by . This book was released on 1992 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Hubeyb Gurdogan
Publisher :
ISBN 13 :
Total Pages : 90 pages
Book Rating : 4.:/5 (133 download)
Download or read book Eigenvector Shrinkage for Estimating Covariance Matrices written by Hubeyb Gurdogan and published by . This book was released on 2021 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: Portfolio managers faced with limited sample sizes must use factor models to estimate the covariance matrix of a high-dimensional returns vector. For the simplest one-factor market model, success rests on the quality of the estimated leading eigenvector ``beta". When only the returns themselves are observed, the practitioner has available the ``PCA" estimate equal to the leading eigenvector of the sample covariance matrix. It is documented that this estimator performs poorly in various ways. To address this problem in the high-dimension, limited sample size asymptotic regime and in the context of estimating the minimum variance portfolio, Goldberg, Papanicolau, and Shkolnik (\cite{goldberg2018}) developed a shrinkage method (the ``GPS estimator") that improves the PCA estimator of beta by shrinking it toward the target unit vector$q = (1,\dots, 1)/\sqrt{p} \in \bR^p$. We investigate what allows this shrinkage to present improvements. We reveal, incorporating the information that $beta$'s are finitely dispersed in the asymptotic regime, leverages the unit vector $q$ as a relevant target vector. In fact, the GPS estimator were using a systematic information inherited by the $beta$'s of the market. This invokes the question of how to pick a favorable shrinkage target in the possible settings that there is further information which promotes more unit vectors like $q$. In that regard, we continue their work to develop a more general framework of shrinkage targets that allows the practitioner to make use of further information to improve the estimator. Examples include sector separation of stock betas, and recent information from prior estimates. We prove some precise statements and illustrate the resulting improvements over the GPS estimator with some numerical experiments. Moreover, we perform an in depth analysis of the asymptotics of the optimization bias term $\mathcal{E}_p$, the major driver of various error metrics of the estimators of $\beta$'s as pointed out in \cite{goldberg2018}, associated wit the GPS estimator
Author : Mohsen Pourahmadi
Publisher : John Wiley & Sons
ISBN 13 : 1118573668
Total Pages : 204 pages
Book Rating : 4.1/5 (185 download)
Download or read book High-Dimensional Covariance Estimation written by Mohsen Pourahmadi and published by John Wiley & Sons. This book was released on 2013-05-28 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Methods for estimating sparse and large covariance matrices Covariance and correlation matrices play fundamental roles in every aspect of the analysis of multivariate data collected from a variety of fields including business and economics, health care, engineering, and environmental and physical sciences. High-Dimensional Covariance Estimation provides accessible and comprehensive coverage of the classical and modern approaches for estimating covariance matrices as well as their applications to the rapidly developing areas lying at the intersection of statistics and machine learning. Recently, the classical sample covariance methodologies have been modified and improved upon to meet the needs of statisticians and researchers dealing with large correlated datasets. High-Dimensional Covariance Estimation focuses on the methodologies based on shrinkage, thresholding, and penalized likelihood with applications to Gaussian graphical models, prediction, and mean-variance portfolio management. The book relies heavily on regression-based ideas and interpretations to connect and unify many existing methods and algorithms for the task. High-Dimensional Covariance Estimation features chapters on: Data, Sparsity, and Regularization Regularizing the Eigenstructure Banding, Tapering, and Thresholding Covariance Matrices Sparse Gaussian Graphical Models Multivariate Regression The book is an ideal resource for researchers in statistics, mathematics, business and economics, computer sciences, and engineering, as well as a useful text or supplement for graduate-level courses in multivariate analysis, covariance estimation, statistical learning, and high-dimensional data analysis.
Author : Jianfeng Yao
Publisher : Cambridge University Press
ISBN 13 : 9781107065178
Total Pages : 0 pages
Book Rating : 4.0/5 (651 download)
Download or read book Large Sample Covariance Matrices and High-Dimensional Data Analysis written by Jianfeng Yao and published by Cambridge University Press. This book was released on 2015-03-26 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has long been observed that several well-known methods in multivariate analysis become inefficient, or even misleading, when the data dimension p is larger than, say, several tens. A seminal example is the well-known inefficiency of Hotelling's T2-test in such cases. This example shows that classical large sample limits may no longer hold for high-dimensional data; statisticians must seek new limiting theorems in these instances. Thus, the theory of random matrices (RMT) serves as a much-needed and welcome alternative framework. Based on the authors' own research, this book provides a first-hand introduction to new high-dimensional statistical methods derived from RMT. The book begins with a detailed introduction to useful tools from RMT, and then presents a series of high-dimensional problems with solutions provided by RMT methods.
Author : Carl-Fredrik Arndt
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (951 download)
Download or read book Studies in Covariance Estimation and Applications in Finance written by Carl-Fredrik Arndt and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis examines estimation of covariance and correlation matrices. More specifically we will in the first part study dynamical properties of the top eigenvalue and eigenvector for sample estimators of covariance and correlation matrices. This is done under the assumption that the top eigenvalue is separated from the others, which is reasonable when the data comes from financial returns. We show exactly how these quantities behave when the true covariance or correlation is stationary and derive theoretical values of related quantities that can be useful when quantifying the amount of non-stationarity for real data. We also validate the results by using Monte-Carlo simulations. A major contribution from the analysis is that it shows how and under which regimes correlation matrices differ from covariance matrices from a dynamic viewpoint. This effect has been observed in financial data, but never explained. In the second part of the thesis we study modifications to covariance estimators that find the optimal estimator within a certain sub-class. This type of estimators is generally known as shrinkage estimators as they modify only eigenvalues of the original estimator. We will do this when the original estimator takes the form A1/2XBXT A1/2, where A and B are matrices and X is a matrix of i.i.d. variables. The analysis is done in the asymptotic limit where both the number of samples and variables approach infinity jointly so that random-matrix theory can be used. Our goal is to find the shrinkage estimator which minimizes expected value of the Frobenius norm between the estimator and the true covariance matrix. To do this we first derive a generalization to the Marchenko-Pastur equation for this class of estimators. This theorem allows us to calculate the asymptotic value of the projection of the sample eigenvectors onto the true covariance matrix. We then show how to use these to find the optimal covariance estimator. At last, we show with simulations that these estimators are close to the optimal bound when used on finite data sets.
Author : Ulf Grenander
Publisher : Courier Corporation
ISBN 13 : 0486462870
Total Pages : 222 pages
Book Rating : 4.4/5 (864 download)
Download or read book Probabilities on Algebraic Structures written by Ulf Grenander and published by Courier Corporation. This book was released on 2008-01-01 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: This systematic approach covers semi-groups, groups, linear vector spaces, and algebra. It states and studies fundamental probabilistic problems for these spaces, focusing on concrete results. 1963 edition.
Author : Hisayuki Tsukuma
Publisher : Springer Nature
ISBN 13 : 9811515964
Total Pages : 119 pages
Book Rating : 4.8/5 (115 download)
Download or read book Shrinkage Estimation for Mean and Covariance Matrices written by Hisayuki Tsukuma and published by Springer Nature. This book was released on 2020-04-16 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained introduction to shrinkage estimation for matrix-variate normal distribution models. More specifically, it presents recent techniques and results in estimation of mean and covariance matrices with a high-dimensional setting that implies singularity of the sample covariance matrix. Such high-dimensional models can be analyzed by using the same arguments as for low-dimensional models, thus yielding a unified approach to both high- and low-dimensional shrinkage estimations. The unified shrinkage approach not only integrates modern and classical shrinkage estimation, but is also required for further development of the field. Beginning with the notion of decision-theoretic estimation, this book explains matrix theory, group invariance, and other mathematical tools for finding better estimators. It also includes examples of shrinkage estimators for improving standard estimators, such as least squares, maximum likelihood, and minimum risk invariant estimators, and discusses the historical background and related topics in decision-theoretic estimation of parameter matrices. This book is useful for researchers and graduate students in various fields requiring data analysis skills as well as in mathematical statistics.
Author : V. S. Pugachev
Publisher : Elsevier
ISBN 13 : 1483190501
Total Pages : 469 pages
Book Rating : 4.4/5 (831 download)
Download or read book Probability Theory and Mathematical Statistics for Engineers written by V. S. Pugachev and published by Elsevier. This book was released on 2014-06-28 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probability Theory and Mathematical Statistics for Engineers focuses on the concepts of probability theory and mathematical statistics for finite-dimensional random variables. The book underscores the probabilities of events, random variables, and numerical characteristics of random variables. Discussions focus on canonical expansions of random vectors, second-order moments of random vectors, generalization of the density concept, entropy of a distribution, direct evaluation of probabilities, and conditional probabilities. The text then examines projections of random vectors and their distributions, including conditional distributions of projections of a random vector, conditional numerical characteristics, and information contained in random variables. The book elaborates on the functions of random variables and estimation of parameters of distributions. Topics include frequency as a probability estimate, estimation of statistical characteristics, estimation of the expectation and covariance matrix of a random vector, and testing the hypotheses on the parameters of distributions. The text then takes a look at estimator theory and estimation of distributions. The book is a vital source of data for students, engineers, postgraduates of applied mathematics, and other institutes of higher technical education.
Author : S.E Ahmed
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (137 download)
Download or read book Large-Sample Estimation Strategies for Eigenvalues of a Wishart Matrix written by S.E Ahmed and published by . This book was released on 1998 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.
Author : Bradley Efron
Publisher :
ISBN 13 :
Total Pages : 21 pages
Book Rating : 4.:/5 (17 download)
Download or read book Multivariate Empirical Bayes and Estimation of Covariance Matrices written by Bradley Efron and published by . This book was released on 1974 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of estimating a covariance matrix in the standard multivariate normal situation is considered. The loss function is one obtained naturally from the problem of estimating several normal mean vectors in an empirical Bayes Situation. Estimators which dominate any constant multiple of the sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as the James-Stein estimator for a mean vector shrinks the maximum likelihood estimators toward a common value.
Author : Erin Hilary Arruda
Publisher :
ISBN 13 :
Total Pages : 128 pages
Book Rating : 4.:/5 (17 download)
Download or read book Applications of Regularization to SEM: Shrinking Eigenvalues to Improve Stability of Covariance Matrices written by Erin Hilary Arruda and published by . This book was released on 2017 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: Estimation methods employed in Structural Equation Modeling (SEM) depend on asymptotic theory. When assumptions are violated (e.g., sample sizes are not especially large relative to the number of variables), methods break down, and conclusions are dubious. It has been suggested that ill-conditioned matrices contribute to poor performance (Huang & Bentler, 2015; Yuan & Bentler, 2017). In the present investigation, a Maximum a Posteriori (MAP) estimator was proposed and implemented for two matrices common to SEM to address conditioning: the sample covariance matrix and the asymptotic covariance matrix based on fourth order moments. This MAP estimator improves the condition of matrices by pushing down (pulling up) the over (under) estimated sample eigenvalues of poorly conditioned matrices, and better-conditioned matrices were expected to improve solution propriety as well as global model fit. Three differing implementations were proposed for Generalized Least Squares estimation methods (GLS and ADF) as well as correction methods to Maximum Likelihood solutions. Potential advantages of the approaches were evaluated using three Monte Carlo simulation studies across a wide range of sample sizes and estimation methods. The results reveal overall solution propriety is improved, and regularization when applied directly to weight matrices is more effective than indirect application (i.e., by modifying an input matrix or using correction methods). Moreover, results were dramatically improved for normal theory GLS even at samples sizes as small as N = 60 and greatly improved for ADF/RES methods at samples as small as N = 150. Generally, advantages did not carry over to non-normal conditions. Potential reasons for this result are given as well as prospective solutions. An illustrative example demonstrates the use of regularized GLS with real-world data.
Author : Robb J. Muirhead
Publisher : John Wiley & Sons
ISBN 13 : 0470316705
Total Pages : 706 pages
Book Rating : 4.4/5 (73 download)
Download or read book Aspects of Multivariate Statistical Theory written by Robb J. Muirhead and published by John Wiley & Sons. This book was released on 2009-09-25 with total page 706 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. ". . . the wealth of material on statistics concerning the multivariate normal distribution is quite exceptional. As such it is a very useful source of information for the general statistician and a must for anyone wanting to penetrate deeper into the multivariate field." -Mededelingen van het Wiskundig Genootschap "This book is a comprehensive and clearly written text on multivariate analysis from a theoretical point of view." -The Statistician Aspects of Multivariate Statistical Theory presents a classical mathematical treatment of the techniques, distributions, and inferences based on multivariate normal distribution. Noncentral distribution theory, decision theoretic estimation of the parameters of a multivariate normal distribution, and the uses of spherical and elliptical distributions in multivariate analysis are introduced. Advances in multivariate analysis are discussed, including decision theory and robustness. The book also includes tables of percentage points of many of the standard likelihood statistics used in multivariate statistical procedures. This definitive resource provides in-depth discussion of the multivariate field and serves admirably as both a textbook and reference.
Author : Olivier Ledoit
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (891 download)
Download or read book Spectrum Estimation: a Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions written by Olivier Ledoit and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Zhidong Bai
Publisher : Springer Science & Business Media
ISBN 13 : 1441906614
Total Pages : 560 pages
Book Rating : 4.4/5 (419 download)
Download or read book Spectral Analysis of Large Dimensional Random Matrices written by Zhidong Bai and published by Springer Science & Business Media. This book was released on 2009-12-10 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users. This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.
