Bilinear Forms and Zonal Polynomials

The book deals with bilinear forms in real random vectors and their generalizations as well as zonal polynomials and their applications in handling generalized quadratic and bilinear forms. The book is mostly self-contained. It starts from basic principles and brings the readers to the current research level in these areas. It is developed with detailed proofs and illustrative examples for easy readability and self-study. Several exercises are proposed at the end of the chapters. The complicated topic of zonal polynomials is explained in detail in this book. The book concentrates on the theoretical developments in all the topics covered. Some applications are pointed out but no detailed application to any particular field is attempted. This book can be used as a textbook for a one-semester graduate course on quadratic and bilinear forms and/or on zonal polynomials. It is hoped that this book will be a valuable reference source for graduate students and research workers in the areas of mathematical statistics, quadratic and bilinear forms and their generalizations, zonal polynomials, invariant polynomials and related topics, and will benefit statisticians, mathematicians and other theoretical and applied scientists who use any of the above topics in their areas. Chapter 1 gives the preliminaries needed in later chapters, including some Jacobians of matrix transformations. Chapter 2 is devoted to bilinear forms in Gaussian real ran­ dom vectors, their properties, and techniques specially developed to deal with bilinear forms where the standard methods for handling quadratic forms become complicated.

1 Preliminaries.- 1.0 Introduction.- 1.1 Jacobians of Matrix Transformations.- 1.1a Some Frequently Used Jacobians in the Real Case.- 1.2 Singular and Nonsingular Normal Distributions.- 1.2a Normal Distribution in the Real Case.- 1.2b The Moment Generating Function for the Real Normal Distribution.- 1.3 Quadratic Forms in Normal Variables.- 1.3a Representations of a Quadratic Form.- 1.3b Representations of the m. g. f. of a Quadratic Expression.- 1.4 Matrix-variate Gamma and Beta Functions.- 1.4a Matrix-variate Gamma, Real Case.- 1.4b Matrix-variate Gamma Density, Real Case.- 1.4c The m. g. f. of a Matrix-variate Real Gamma Variable.- 1.4d Matrix-variate Beta, Real Case.- 1.5 Hypergeometric Series, Real Case.- 2 Quadratic and Bilinear Forms in Normal Vectors.- 2.0 Introduction.- 2.1 Various Representations.- 2.2 Density of a Gamma Difference.- 2.2a Some Particular Cases.- 2.3 Noncentral Gamma Difference.- 2.4 Moments and Cumulants of Bilinear Forms.- 2.4a Joint Moments and Cumulants of Quadratic and Bilinear Forms.- 2.4a.1 The First Few Joint Cumulants of a Bilinear Form and a Quadratic Form in the Nonsingular Normal Case.- 2.4a.2 The First Few Cumulants of a Quadratic Form in the Nonsingular Normal Case.- 2.4a.3 The First Few Cumulants of a Bilinear Form in the Nonsingular Normal Case.- 2.4b Joint Cumulants of Bilinear Forms.- 2.4b.1 The First Few Joint Cumulants of of Two Bilinear Forms.- 2.4c Moments and Cumulants in the Singular Normal Case.- 2.4c.1 The First Few Cumulants of a Bilinear Form in the Singular and Nonsingular Normal Cases.- 2.4d Cumulants of Bilinear Expressions.- 2. 4d. l Some Special Cases of Bilinear Expressions.- 2.5 Laplacianness of Bilinear Forms.- 2.5a Quadratic and Bilinear Forms in the Nonsingular Normal Case.- 2.5b NS Conditions for the Noncorrelated Normal Case.- 2.5c Quadratic and Bilinear Forms in the Singular Normal Case.- 2.5d Noncorrelated Singular Normal Case.- 2.5e The NS Conditions for a Quadratic Form to be NGL.- 2.6 Generalizations to Bilinear and Quadratic Expressions.- 2.6a Bilinear and Quadratic Expressions in the Nonsingular Normal Case.- 2.6b Bilinear and Quadratic Expressions in the Singular Normal Case.- 2.7 Independence of Bilinear and Quadratic Expressions.- 2.7a Independence of a Bilinear and a Quadratic Form.- 2.7b Independence of Two Bilinear Forms.- 2.7c Independence of Quadratic Expressions: Nonsingular Normal Case.- 2.7d Independence in the Singular Normal Case.- 2.8 Bilinear Forms and Noncentral Gamma Differences.- 2.8a Bilinear Forms in the Equicorrelated Case.- 2.8b Noncentral Case.- 2.9 Rectangular Matrices.- 2.9a Matrix-variate Laplacian.- 2.9b The Density of S2i.- 2.9c A Particular Case.- Exercises.- 3 Quadratic and Bilinear Forms in Elliptically Contoured Distributions.- 3.0 Introduction.- 3.1 De£nitions and Basic Results.- 3.2 Moments of Quadratic Forms.- 3.3 The Distribution of Quadratic Forms.- 3.4 Noncentral Distribution.- 3.5 Quadratic Forms in Random Matrices.- 3.6 Quadratic Forms of Random Idempotent Matrices.- 3.7 Cochran’s Theorem.- 3.8 Test Statistics for Elliptically Contoured Distributions.- Sample Correlation Coefficient.- Likelihood Ratio Criteria.- Exercises.- 4 Zonal Polynomials.- 4.0 Introduction.- 4.1 Wishart Distribution.- 4.2 Symmetric Polynomials.- 4.3 Zonal Polynomials.- 4.4 Laplace Transform and Hypergeometric Function.- 4.5 Binomial Coefficients.- 4.6 Some Special Functions.- Exercises.- Table 4.3.2(a).- Table 4.3.2(b).- Table 4.4.1.- 5 Generalized Quadratic Forms.- 5.0 Introduction.- 5.1 A Representation of the Distribution of a Generalized Quadratic Form.- 5.2 An Alternate Representation.- 5.3 The Distribution of the Latent Roots of a Quadratic Form.- 5.4 Distributions of Some Functions of XAX’.- 5.5 Generalized Hotelling’s T02.- 5.6 Anderson’s Linear Discriminant Function.- 5.7 Multivariate Calibration.- 5.8 Asymptotic Expansions of the Distribution of a Quadratic Form.- Exercises.- Table 5.6.1.- Table 5.6.2.- Appendix Invariant Polynomials.- Appendix A. l Representation of a Group.- Appendix A.2 Integration of the Representation Matrix over the Orthogonal Group.- Glossary of Symbols.- Author Index.