A comprehensive examination of high-dimensional analysis of
multivariate methods and their real-world applications
Multivariate 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 comparison
Each 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.
Autorentext
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
A comprehensive examination of high-dimensional analysis of multivariate methods and their real-world applications
Multivariate 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 comparison
Each 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.
Inhalt
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 T2 and Lambda Statistics.
3.1 Hotelling's T2 and Lambda Statistics.
3.1.1 Distribution of the T2 Statistic.
3.1.2 Decomposition of T2 and D2.
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 T2Statistic.
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 R…