## Multivariate Statistics: High-dimensional and Large-sample Approximations

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. Emphasizing that high-dimensional asymptotic distribution can be used for a large range of samples and dimensions to achieve high levels of accuracy, this timely text provides approximation formulas, actual applications, thorough analysis of the real data, and solutions to each problem that are useful to both practical and theoretical statisticians as well as graduate students.

### Biography

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 numerous journal articles on probability distributions.

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.

3.3.1 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 Multivariate 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.

9.4.3 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.

12.2.2 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.

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