Multivariate statistics : highdimensional and largesample approximations
 Responsibility
 Yasunori Fujikoshi, Vladimir V. Ulyanov, Ryoichi Shimizu.
 Language
 English.
 Imprint
 Hoboken, N.J. : Wiley, c2010.
 Physical description
 xviii, 533 p. : ill ; 25 cm.
 Series
 Wiley series in probability and statistics.
Access
Creators/Contributors
 Author/Creator
 Fujikoshi, Yasunori, 1942
 Contributor
 Ulyanov, Vladimir V., 1953
 Shimizu, Ryoichi, 1931
Contents/Summary
 Bibliography
 Includes bibliographical references (p. [513]526) and index.
 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 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 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 T2Statistic. 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 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 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 S1 = S2. 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 L1Norm. 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 T²0 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 . 16.2.3 Confidence Interval for . 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.
 (source: Nielsen Book Data)
 Publisher's Summary
 Written by wellknown, awardwinning authors, this is the first book to focus on highdimensional data analysis while presenting realworld applications and research material. Emphasizing that highdimensional 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2010
 Series
 Wiley series in probability and statistics
 ISBN
 9780470411698 (cloth)
 0470411694 (cloth)