Multivariate Statistics : HighDimensional and LargeSample ApproximationsISBN: 9780470411698
512 pages
January 2010

Multivariate Statistics: HighDimensional and LargeSample 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 highdimensional and largescale 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 highdimensional 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:
 Highdimensional approximations of various statistics
 Highdimensional 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 realworld 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.
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 LambdaStatistic.
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 tStatistic.
5.3.3 Asymptotic Expansions.
5.4 Statistics with a Class of Moments.
5.4.1 LargeSample Expansions.
5.4.2 HighDimensional Expansions.
5.5 Perturbation Method.
5.6 CornishFisher Expansions.
5.6.1 Expansion Formulas.
5.6.2 Validity of CornishFisher Expansions.
5.7 Transformations for Improved Approximations.
5.8 Bootstrap Approximations.
5.9 HighDimensional 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 OneWay Analysis of Variance.
6.2 Multivariate TwoWay Analysis of Variance.
6.3 MANOVA Tests.
6.3.1 Test Criteria.
6.3.2 LargeSample Approximations.
6.3.3 Comparison of Powers.
6.3.4 HighDimensional 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 LargeSample Case.
6.5.3 HighDimensional Case.
6.6 Tests for Dimensionality.
6.6.1 Three Test Criteria.
6.6.2 LargeSample and HighDimensional Asymptotics.
6.7 HighDimensional 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 HighDimensional 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 HighDimensional Tests.
8.2 Sphericity.
8.2.1 Likelihood Ratio Tests and Moments.
8.2.2 Asymptotic Expansions.
8.2.3 HighDimensional 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 HighDimensional Tests.
8.5 Tests for Equality of Covariance Matrices.
8.5.1 Likelihood Ratio Test and Moments.
8.5.2 Asymptotic Expansions.
8.5.3 HighDimensional Tests.
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 S_{1} = S_{2}.
9.2.2 Case of Several Normal Populations.
9.3 Probability of Misclassifications.
9.3.1 WRule.
9.3.2 ZRule.
9.3.3 HighDimensional Asymptotic Results.
9.4 Canonical Discriminant Analysis.
9.4.1 Canonical Discriminant 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 HighDimensional Approach.
9.6.1 Penalized Discriminant Analysis.
9.6.2 Other Approaches.
Problems.
10 Principal Component Analysis.
10.1 Definition of Principal Components.
10.2 Optimality of Principal Components.
10.3 Sample Principal Components.
10.4 MLEs of the Characteristic Roots and Vectors.
10.5 Distributions of the Characteristic Roots.
10.5.1 Exact Distribution.
10.5.2 LargeSample Case.
10.5.3 HighDimensional Case.
10.6 Model Selection Approach for Covariance Structures.
10.6.1 General Approach.
10.6.2 Models for Equality of the Smaller Roots.
10.6.3 Selecting a Subset of Original Variables.
10.7 Methods Related to Principal Components.
10.7.1 FixedEffect Principal Component Model.
10.7.2 RandomEffect Principal Components Model.
Problems.
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 LargeSample Asymptotic Distributions.
11.3.3 HighDimensional Asymptotic Distributions.
11.3.4 Fisher's zTransformation.
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.
Problems.
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 Derivation of CAIC.
12.5.3 Extended Growth Curve Model.
Problems.
13 Approximation to the ScaleMixted Distributions.
13.1 Introduction.
13.1.1 Simple Example: Student's tDistribution.
13.1.2 Improving the Approximation.
13.2 Error Bounds Evaluated in SupNorm.
13.2.1 General Theory.
13.2.2 ScaleMixed Normal.
13.2.3 ScaleMixed Gamma.
13.3 Error Bounds Evaluated in L_{1}Norm.
13.3.1 Some Basic Results.
13.3.2 ScaleMixed Normal Density.
13.3.3 ScaleMixed Gamma Density.
13.3.4 ScaleMixed Chisquare Density.
13.4 Multivariate Scale Mixtures.
13.4.1 General Theory.
13.4.2 Normal Case.
13.4.3 Gamma Case.
Problems.
14 Approximation to Some Related Distributions.
14.1 Location and Scale Mixtures.
14.2 Maximum of Multivariate Variables.
14.2.1 Distribution of the Maximum Component of a Multivariate Variable.
14.2.2 Multivariate tDistribution.
14.2.3 Multivariate FDistribution.
14.3 Scale Mixtures of the FDistribution.
14.4 NonUniform Error Bounds.
14.5 Method of Characteristic Functions.
Problems.
15 Error Bounds for Approximations of 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.
Problems.
16 Error Bounds for Approximations to Some Other Statistics.
16.1 Linear Discriminant Function.
16.1.1 Representation as Location and Scale Mixture.
16.1.2 LargeSample Approximations.
16.1.3 HighDimensional Approximations.
16.1.4 Some Related Topics.
16.2 Profile Analysis.
16.2.1 Parallelism 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 Error Bounds.
16.3.2 Distribution of the Bilinear Form.
16.4 Generalized Least Squares Estimators.
Problems.
Appendix.
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 Idempotent Matrices.
A.2 Inequalities and MaxMin Problems.
A.3 Jacobians of Transformations.
Bibliography.
Index.
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.
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