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.

• Preface. Glossary of Notation and Abbreviations.1 Multivariate Normal and Related Distributions.1.1 Random Vectors.1.2 Multivariate Normal Distribution.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.3.2 Lambda-Statistic.3.3 Test for Additional Information.Problems.4 Correlation Coefficients.4.1 Ordinary Correlation Coefficients.4.2 Multiple Correlation Coefficient.4.3 Partial Correlation.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.5.4 Statistics with a Class of Moments.5.5 Perturbation Method.5.6 Cornish-Fisher Expansions.5.7 Transformations for Improved Approximations.5.8 Bootstrap Approximations.5.9 High-Dimensional Approximations.Problems.6 MANOVA Models.6.1 Multivariate One-Way Analysis of Variance.6.2 Multivariate Two-Way Analysis of Variance.6.3 MANOVA Tests.6.4 Approximations Under Nonnormality.6.5 Distributions of Characteristic Roots.6.6 Tests for Dimensionality.6.7 High-Dimensional Tests.Problems.7 Multivariate Regression.7.1 Multivariate Linear Regression Model.7.2 Statistical Inference.7.3 Selection of Variables.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 High-Dimensional Tests for Covariance Matrices.8.1 Specified Covariance Matrix.8.2 Sphericity.8.3 Intraclass Covariance Structure.8.4 Test for Independence.8.5 Tests for Equality of Covariance Matrices.Problems.9 Discriminant Analysis.9.1 Classification Rules for Known Distributions.9.2 Sample Classification Rules for Normal Populations.9.3 Probability of Misclassifications.9.4 Canonical Discriminant Analysis.9.5 Regression Approach.9.6 High-Dimensional Approach.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.10.6 Model Selection Approach for Covariance Structures.10.7 Methods Related to Principal Components.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.11.4 Inference for Dimensionality.11.5 Selection of Variables.Problems.12 Growth Curve Analysis.12.1 Growth Curve Model.12.2 Statistical Inference: One Group.12.3 Statistical Methods: Several Groups.12.4 Derivation of Statistical Inference.12.5 Model Selection.Problems.13 Approximation to the Scale-Mixted Distributions.13.1 Introduction.13.2 Error Bounds Evaluated in Sup-Norm.13.3 Error Bounds Evaluated in L1-Norm.13.4 Multivariate Scale Mixtures.Problems.14 Approximation to Some Related Distributions.14.1 Location and Scale Mixtures.14.2 Maximum of Multivariate Variables.14.3 Scale Mixtures of the F-Distribution.14.4 Non-Uniform 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.Problems. 16 Error Bounds for Approximations to Some Other Statistics.16.1 Linear Discriminant Function.16.2 Profile Analysis.16.3 Estimators in the Growth Curve Model.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 Max-Min Problems.A.3 Jacobians of Transformations.Bibliography.Index.

"The book is designed for readers interested in multivariate analysis with a good background in matrix algebra, mathematical statistical inference and probability theory. Its contents are, in general, well organised and the intuitive ideas behind the different multivariate methods, the asymptotic expansion techniques and the calculation of error bounds using scale mixtures, are well expressed . The mathematical proofs are well presented and selected and I have found the mathematical appendices to be very useful as guides to following the proofs." (Mathematical Reviews, 2011)

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.

• Preface. Glossary of Notation and Abbreviations.1 Multivariate Normal and Related Distributions.1.1 Random Vectors.1.2 Multivariate Normal Distribution.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.3.2 Lambda-Statistic.3.3 Test for Additional Information.Problems.4 Correlation Coefficients.4.1 Ordinary Correlation Coefficients.4.2 Multiple Correlation Coefficient.4.3 Partial Correlation.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.5.4 Statistics with a Class of Moments.5.5 Perturbation Method.5.6 Cornish-Fisher Expansions.5.7 Transformations for Improved Approximations.5.8 Bootstrap Approximations.5.9 High-Dimensional Approximations.Problems.6 MANOVA Models.6.1 Multivariate One-Way Analysis of Variance.6.2 Multivariate Two-Way Analysis of Variance.6.3 MANOVA Tests.6.4 Approximations Under Nonnormality.6.5 Distributions of Characteristic Roots.6.6 Tests for Dimensionality.6.7 High-Dimensional Tests.Problems.7 Multivariate Regression.7.1 Multivariate Linear Regression Model.7.2 Statistical Inference.7.3 Selection of Variables.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 High-Dimensional Tests for Covariance Matrices.8.1 Specified Covariance Matrix.8.2 Sphericity.8.3 Intraclass Covariance Structure.8.4 Test for Independence.8.5 Tests for Equality of Covariance Matrices.Problems.9 Discriminant Analysis.9.1 Classification Rules for Known Distributions.9.2 Sample Classification Rules for Normal Populations.9.3 Probability of Misclassifications.9.4 Canonical Discriminant Analysis.9.5 Regression Approach.9.6 High-Dimensional Approach.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.10.6 Model Selection Approach for Covariance Structures.10.7 Methods Related to Principal Components.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.11.4 Inference for Dimensionality.11.5 Selection of Variables.Problems.12 Growth Curve Analysis.12.1 Growth Curve Model.12.2 Statistical Inference: One Group.12.3 Statistical Methods: Several Groups.12.4 Derivation of Statistical Inference.12.5 Model Selection.Problems.13 Approximation to the Scale-Mixted Distributions.13.1 Introduction.13.2 Error Bounds Evaluated in Sup-Norm.13.3 Error Bounds Evaluated in L1-Norm.13.4 Multivariate Scale Mixtures.Problems.14 Approximation to Some Related Distributions.14.1 Location and Scale Mixtures.14.2 Maximum of Multivariate Variables.14.3 Scale Mixtures of the F-Distribution.14.4 Non-Uniform 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.Problems. 16 Error Bounds for Approximations to Some Other Statistics.16.1 Linear Discriminant Function.16.2 Profile Analysis.16.3 Estimators in the Growth Curve Model.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 Max-Min Problems.A.3 Jacobians of Transformations.Bibliography.Index.

"The book is designed for readers interested in multivariate analysis with a good background in matrix algebra, mathematical statistical inference and probability theory. Its contents are, in general, well organised and the intuitive ideas behind the different multivariate methods, the asymptotic expansion techniques and the calculation of error bounds using scale mixtures, are well expressed . The mathematical proofs are well presented and selected and I have found the mathematical appendices to be very useful as guides to following the proofs." (Mathematical Reviews, 2011)

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.

• Preface. Glossary of Notation and Abbreviations.1 Multivariate Normal and Related Distributions.1.1 Random Vectors.1.2 Multivariate Normal Distribution.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.3.2 Lambda-Statistic.3.3 Test for Additional Information.Problems.4 Correlation Coefficients.4.1 Ordinary Correlation Coefficients.4.2 Multiple Correlation Coefficient.4.3 Partial Correlation.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.5.4 Statistics with a Class of Moments.5.5 Perturbation Method.5.6 Cornish-Fisher Expansions.5.7 Transformations for Improved Approximations.5.8 Bootstrap Approximations.5.9 High-Dimensional Approximations.Problems.6 MANOVA Models.6.1 Multivariate One-Way Analysis of Variance.6.2 Multivariate Two-Way Analysis of Variance.6.3 MANOVA Tests.6.4 Approximations Under Nonnormality.6.5 Distributions of Characteristic Roots.6.6 Tests for Dimensionality.6.7 High-Dimensional Tests.Problems.7 Multivariate Regression.7.1 Multivariate Linear Regression Model.7.2 Statistical Inference.7.3 Selection of Variables.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 High-Dimensional Tests for Covariance Matrices.8.1 Specified Covariance Matrix.8.2 Sphericity.8.3 Intraclass Covariance Structure.8.4 Test for Independence.8.5 Tests for Equality of Covariance Matrices.Problems.9 Discriminant Analysis.9.1 Classification Rules for Known Distributions.9.2 Sample Classification Rules for Normal Populations.9.3 Probability of Misclassifications.9.4 Canonical Discriminant Analysis.9.5 Regression Approach.9.6 High-Dimensional Approach.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.10.6 Model Selection Approach for Covariance Structures.10.7 Methods Related to Principal Components.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.11.4 Inference for Dimensionality.11.5 Selection of Variables.Problems.12 Growth Curve Analysis.12.1 Growth Curve Model.12.2 Statistical Inference: One Group.12.3 Statistical Methods: Several Groups.12.4 Derivation of Statistical Inference.12.5 Model Selection.Problems.13 Approximation to the Scale-Mixted Distributions.13.1 Introduction.13.2 Error Bounds Evaluated in Sup-Norm.13.3 Error Bounds Evaluated in L1-Norm.13.4 Multivariate Scale Mixtures.Problems.14 Approximation to Some Related Distributions.14.1 Location and Scale Mixtures.14.2 Maximum of Multivariate Variables.14.3 Scale Mixtures of the F-Distribution.14.4 Non-Uniform 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.Problems. 16 Error Bounds for Approximations to Some Other Statistics.16.1 Linear Discriminant Function.16.2 Profile Analysis.16.3 Estimators in the Growth Curve Model.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 Max-Min Problems.A.3 Jacobians of Transformations.Bibliography.Index.

"The book is designed for readers interested in multivariate analysis with a good background in matrix algebra, mathematical statistical inference and probability theory. Its contents are, in general, well organised and the intuitive ideas behind the different multivariate methods, the asymptotic expansion techniques and the calculation of error bounds using scale mixtures, are well expressed . The mathematical proofs are well presented and selected and I have found the mathematical appendices to be very useful as guides to following the proofs." (Mathematical Reviews, 2011)