Multivariate Statistics: High–Dimensional and Large–Sample Approximations

Wiley Series In Probability And Statistics
Multivariate Statistics
High–Dimensionaland Large–Sample Approximations
Yasunori Fujikoshi
Vladimir V. Ulyanov
Ryoichi Shimizu
A comprehensive examination ofhigh–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.

Spis treści:
1 Multivariate Normal and Related Distributions.
1.1 Random Vectors.
1.1.1 Mean Vector and Covariance Matrix.
1.1.2 The 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.
1.5 Problems.
2 Wishart Distribution.
2.1 Definition.
2.2 Some Basic Properties.
2.3 Functions of Wishart Matrices.
2.4 Cochran Theorem.
2.5 Asymptotic Distributions.
2.6 Problems.
3 T2– and Lambda–Statistics.
3.1 T2–Statistic.
3.1.1 Distribution of T2–Statistic.
3.1.2 Decomposition of T2 or 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.
3.4 Problems.
4 Correlation Coefficients.
4.1 Ordinary Correlation Coefficients 65.
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.
4.4 Problems.
5 Asymptotic Expansions for Multivar iate Basic Statistics.
5.1 Edgeworth Expansion and Its Validity.
5.2 The Sample Mean Vector and Covariance Matrix.
5.3 T2–Statistic.
5.3.1 Outlines of Two Methods.
5.3.2 Multivariate t–Statistic.
5.3.3 Asmptotic Expansionz.
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.
5.10 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 Tests158.
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.
6.8 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 Cp 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 Problems201.
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 Asymptotitc 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 Struture.
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 Test for Equality of Several Covariance Matrices.
8.5.1 Likelihood Ratio Test and Moments.
8.5.2 Asymptotic Expansions.
8.5.3 High–dimensional Tests.
8.6 Problems.
9 Discriminant Analysis.
9.1 Classification Rules for Known Distributions.
9.2 Sample Classification Rules for Normal Populations.
9.2.1 Two Normal Populations with Σ1 = Σ2.
9.2.2 Case of Several Normal Populations.
9.3 Probabilities of Misclassications.
9.3.1 W–Rule.
9.3.2 Z–Rule.
9.3.3 High–Dimensional Asymptotic Results.
9.4 Canonical Discriminant Analysis.
9.4.1 Canonical Dicriminant Method.
9.4.2 Test for Additional Information.
9.4.3 Selection of Variables.
9.4.4 Estimation of Dimensionality.
9.5 Regression Approach.
9.6 High–Dimensional Approach.
9.7 Problems.
10 Principal Component Analysis.
10.1 Definition of Principal Components.
10.2 Optimality of Principal Components.
10.3 Sample Principal Components.
10.4 Distributions of the Characteristic Roots.
10.5 Model Selection Approach for Covariance Structures.
10.5.1 A General Approach.
10.5.2 Models for Equality of the Smaller Roots.
10.6 Methods Related to Principal Components.
10.6.1 Fixed principal component model.
10.6.2 Random Effect Principal Components Model.
10.7 Problem.
11 Canonical Correlation Analysis.
11.1 Definition of Population Canonical Correlations and Variables.
11.2 Sample Canonical Correlations.
11.3 Distributions of Canonical Correlations.
11.3.1 Distributional Reduction.
11.3.2 Large–Sample Asymptotic.
11.3.3 High–Dimensional Asymptotic Distributions.
11.3.4 Fisher′s z–Transformation.
11.4 Inference for Dimensionality.
11.4.1 Test of Dimensionality.
11.4.2 Estimation of Dimensionality.
11.5 Selection of Variables.
11.5.1 Test for Redundancy.
11.5.2 Selection of Variables.
11.6 Problem.
12 Growth Curve Analysis.
12.1 Growth Curve Model.
12.2 Statistical Inference – One Group.
12.2.1 Test for Adequacy.
12.2.2 Estimation and Test.
12.2.3 Confidence Intervals.
12.3 Statistical Methods – Several Groups.
12.4 Derivation of Statistical Inference.
12.4.1 A General Multivariate Linear Model.
12.4.2 Estimation.
12.4.3 LR Tests for General Linear Hypotheses.
12.4.4 Confidence Intervals.
12.5 Model Selection.
12.5.1 AIC and CAIC.
12.5.2 Deivation of CAIC.
12.5.3 An Extended Growth Curve Model.
13 Theory of Approximation to the Distribution of Scale Mixture.
13.1 Introduction.
13.2 Approximating Scale Mixture of Distributions.
13.2.1 General Theory.
13.2.2 Scale Mixed Normal.
13.2.3 Scale Mixed Gamma.
13.3 Error Bounds Evaluated in L1–Norm.
13.3.1 Some Basic Results.
13.3.2 Scale Mixed Normal Density.
13.3.3 Scale Mixed Gamma Density.
13.3.4 Scale Mixture of χ2(q).
13.4 Multivariate Scale Mixtures.
13.4.1 General Theory.
13.4.2 Normal Case.
13.4.3 Gamma Case.
13.4.4 Problems.
14 Basic Theory of Approximation to Some Related Distributions.
14.1 Location and Scale Mixtures.
14.2 The maximum of Multivariate t– and F–variables.
14.3 Scale Mixtures of the F–distribution.
14.4 Non–Uniform Error Bounds.
14.5 Methof of Characterixtic Functions.
15 Error Bounds for Approximations of Some Multivariate Tests.
15.1 Multivariate Scale Mixture and MANOVA Tests.
15.2 A Function of Multivariate Scale Mixture.
15.3 Hotelling′s T2 0 –Statistic.
15.4 Wilk′s Lambda Distribution.
15.4.1 Univariate Case.
15.4.2 Multivariate Case.
16 Error Bounds for Approximations of Some Other Statistics.
16.1 Linear Discriminant Function.
16.1.1 Representation as Location and Scale Mixture.
16.1.2 Large Sample Approximations.
16.1.3 High Dimensional Approximations.
16.1.4 Some Related Topics.
16.2 Profile Analysis.
16.2.1 Prallelism Model and MLE.
16.2.2 Distributions of ^γ.
16.2.3 Confidence Interval for γ.
16.3 Estimators in the Growth Curve Model.
16.3.1 Growth Curve Model — Error Bounds.
16.3.2 Distribution of the Bi–Linear Form.
16.4 Generalized least squares estimators.
16.5 Problems.
A Appendix: Some Results on Matrices.
A.1 Some Results on Matrices.
A.1.1 Determinants and Inverse Matrices.
A.1.2 Characteristic Roots and Vectors.
A.1.3 Matrix Factorizations.
A.1.4 Idenpotent Matrices.
A.2 Inequalities and Max–min Problems.
A.3 Jacobians of transformations.
Bibliography.

Nota biograficzna:

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 Mathe–matics (Japan) and is the author of numer ous journal articles on probability distributions.

Okładka tylna:

Wiley Series In Probability And Statistics
Multivariate Statistics
High–Dimensionaland Large–Sample Approximations
Yasunori Fujikoshi
Vladimir V. Ulyanov
Ryoichi Shimizu
A comprehensive examination ofhigh–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 foun d 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.