Multivariate Statistics - High-Dimensional and Large-Sample Approximations

Informationen zum Autor Yasunori Fujikoshi, DSc, is Professor Emeritus at Hiroshima University (Japan) and Visiting Professor in the Department of Mathematics at Chuo University (Japan). He has authored over 150 journal articles in the area of multivariate analysis. Vladimir V. Ulyanov, DSc, is Professor in the Department of Mathematical Statistics at Moscow State University (Russia) and is the author of nearly fifty journal articles in his areas of research interest, which include weak limit theorems, probability measures on topological spaces, and Gaussian processes. Ryoichi Shimizu, DSc, is Professor Emeritus at the Institute of Statistical Mathematics (Japan) and is the author of numerous journal articles on probability distributions. Klappentext Wiley Series In Probability And StatisticsMultivariate StatisticsHigh-Dimensionaland Large-Sample ApproximationsYasunori FujikoshiVladimir V. UlyanovRyoichi ShimizuA comprehensive examination ofhigh-dimensional analysis of multivariate methods and their real-world applicationsMultivariate Statistics: High-Dimensional and Large-Sample Approximations is the first book of its kind to explore how classical multivariate methods can be revised and used in place of conventional statistical tools. Written by prominent researchers in the field, the book focuses on high-dimensional and large-scale approximations and details the many basic multivariate methods used to achieve high levels of accuracy.The authors begin with a fundamental presentation of the basic tools and exact distributional results of multivariate statistics, and, in addition, the derivations of most distributional results are provided. Statistical methods for high-dimensional data, such as curve data, spectra, images, and DNA microarrays, are discussed. Bootstrap approximations from a methodological point of view, theoretical accuracies in MANOVA tests, and model selection criteria are also presented. Subsequent chapters feature additional topical coverage including:* High-dimensional approximations of various statistics* High-dimensional statistical methods* Approximations with computable error bound* Selection of variables based on model selection approach* Statistics with error bounds and their appearance in discriminant analysis, growth curve models, generalized linear models, profile analysis, and multiple comparisonEach chapter provides real-world applications and thorough analyses of the real data. In addition, approximation formulas found throughout the book are a useful tool for both practical and theoretical statisticians, and basic results on exact distributions in multivariate analysis are included in a comprehensive, yet accessible, format.Multivariate Statistics is an excellent book for courses on probability theory in statistics at the graduate level. It is also an essential reference for both practical and theoretical statisticians who are interested in multivariate analysis and who would benefit from learning the applications of analytical probabilistic methods in statistics. Zusammenfassung Written by well-known, award-winning authors, this is the first book to focus on high-dimensional data analysis while presenting real-world applications and research material. Inhaltsverzeichnis Preface. Glossary of Notation and Abbreviations. 1 Multivariate Normal and Related Distributions. 1.1 Random Vectors. 1.1.1 Mean Vector and Covariance Matrix. 1.1.2 Characteristic Function and Distribution. 1.2 Multivariate Normal Distribution. 1.2.1 Bivariate Normal Distribution. 1.2.2 Definition. 1.2.3 Some Properties. 1.3 Spherical and Elliptical Distributions. 1.4 Multivariate Cumulants. Problems. 2 Wishart Distribution. 2.1 Definition. 2.2 Some Basic Properties. 2.3 Functions of Wishart Matrices. 2...

List of contents

Preface.
Glossary of Notation and Abbreviations.

1 Multivariate Normal and Related Distributions.

1.1 Random Vectors.

1.1.1 Mean Vector and Covariance Matrix.

1.1.2 Characteristic Function and Distribution.

1.2 Multivariate Normal Distribution.

1.2.1 Bivariate Normal Distribution.

1.2.2 Definition.

1.2.3 Some Properties.

1.3 Spherical and Elliptical Distributions.

1.4 Multivariate Cumulants.

Problems.

2 Wishart Distribution.

2.1 Definition.

2.2 Some Basic Properties.

2.3 Functions of Wishart Matrices.

2.4 Cochran s Theorem.

2.5 Asymptotic Distributions.

Problems.

3 Hotelling s T 2 and Lambda Statistics.

3.1 Hotelling s T 2 and Lambda Statistics.

3.1.1 Distribution of the T 2 Statistic.

3.1.2 Decomposition of T 2 and D 2 .

3.2 Lambda-Statistic.

3.2.1 Motivation of Lambda Statistic.

3.2.2 Distribution of Lambda Statistic.

3.3 Test for Additional Information.

3.3.1 Decomposition of Lambda Statistic.

Problems.

4 Correlation Coefficients.

4.1 Ordinary Correlation Coefficients.

4.1.1 Population Correlation.

4.1.2 Sample Correlation.

4.2 Multiple Correlation Coefficient.

4.2.1 Population Multiple Correlation.

4.2.2 Sample Multiple Correlation.

4.3 Partial Correlation.

4.3.1 Population Partial Correlation.

4.3.2 Sample Partial Correlation.

4.3.3 Covariance Selection Model.

Problems.

5 Asymptotic Expansions for Multivariate Basic Statistics.

5.1 Edgeworth Expansion and its Validity.

5.2 The Sample Mean Vector and Covariance Matrix.

5.3 T 2 Statistic.

5.3.1 Outlines of Two Methods.

5.3.2 Multivariate t-Statistic.

5.3.3 Asymptotic Expansions.

5.4 Statistics with a Class of Moments.

5.4.1 Large-Sample Expansions.

5.4.2 High-Dimensional Expansions.

5.5 Perturbation Method.

5.6 Cornish-Fisher Expansions.

5.6.1 Expansion Formulas.

5.6.2 Validity of Cornish-Fisher Expansions.

5.7 Transformations for Improved Approximations.

5.8 Bootstrap Approximations.

5.9 High-Dimensional Approximations.

5.9.1 Limiting Spectral Distribution.

5.9.2 Central Limit Theorem.

5.9.3 Martingale Limit Theorem.

5.9.4 Geometric Representation.

Problems.

6 MANOVA Models.

6.1 Multivariate One-Way Analysis of Variance.6.2 Multivariate Two-Way Analysis of Variance.

6.3 MANOVA Tests.

6.3.1 Test Criteria.

6.3.2 Large-Sample Approximations.

6.3.3 Comparison of Powers.

6.3.4 High-Dimensional Approximations.

6.4 Approximations Under Nonnormality.

6.4.1 Asymptotic Expansions.

6.4.2 Bootstrap Tests.

6.5 Distributions of Characteristic Roots.

6.5.1 Exact Distributions.

6.5.2 Large-Sample Case.

6.5.3 High-Dimensional Case.

6.6 Tests for Dimensionality.

6.6.1 Three Test Criteria.

6.6.2 Large-Sample and High-Dimensional Asymptotics.

6.7 High-Dimensional Tests.

Problems.

7 Multivariate Regression.

7.1 Multivariate Linear Regression Model.

7.2 Statistical Inference.

7.3 Selection of Variables.

7.3.1 Stepwise Procedure.

7.3.2 C p Criterion.

7.3.3 AIC Criterion.

7.3.4 Numerical Example.

7.4 Principal Component Regression.

7.5 Selection of Response Variables.

7.6 General Linear Hypotheses and Confidence Intervals.

7.7 Penalized Regression Models.

Problems.

8 Classical and High-Dimensional Tests for Covariance Matrices.

8.1 Specified Covariance Matrix.

8.1.1 Likelihood Ratio Test and Moments.

8.1.2 Asymptotic Expansions.

8.1.3 High-Dimensional Tests.

8.2 Sphericity.

8.2.1 Likelihood Ratio Tests and Moments.

8.2.2 Asymptotic Expansions.

8.2.3 High-Dimensional Tests.

8.3 Intraclass Covariance Structure.

8.3.1 Likelihood Ratio Tests and Moments.

8.3.2 Asymptotic Expansions.

8.3.3 Numerical Accuracy.

8.4 Test for Independence.

8.4.1 Likelihood Ratio Tests and Moments.

8.4.2 Asymptotic Expansions.

8.4.3 High-Dimensional Tests.

8.5 Tests for Equality of Covariance

Report

"The book is designed for readers interested in multivariate analysis with a good background in matrix algebra, mathematical statistical inference and probability theory. Its contents are, in general, well organised and the intuitive ideas behind the different multivariate methods, the asymptotic expansion techniques and the calculation of error bounds using scale mixtures, are well expressed . The mathematical proofs are well presented and selected and I have found the mathematical appendices to be very useful as guides to following the proofs." (Mathematical Reviews, 2011)

Product details

• Authors: Y Fujikoshi, Yasunor Fujikoshi, Yasunori Fujikoshi, Yasunori (Hiroshima University) Ulyanov Fujikoshi, Yasunori Ulyanov Fujikoshi, Yasunori/ Ulyanov Fujikoshi, R Shimizu, Ryoichi Shimizu, Shimizu Ryoichi, Vladimir Ulyanov, Vladimir V Ulyanov, Vladimir V. Ulyanov, Ulyanov Vladimir V.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 09.02.2010
• EAN: 9780470411698
• ISBN: 978-0-470-41169-8
• No. of pages: 568
• Series: Wiley Series in Probability and Statistics; Wiley Series in Probability an; Wiley Series in Probability and Statistics; Wiley Probability and Statisti
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Statistik, Stichprobe, Statistics, Multivariate Analyse, Multivariate Analysis, Methoden der Daten- u. Stichprobenerhebung, Survey Research Methods & Sampling