Bilinear Forms and Zonal Polynomials

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.- 2.4b Joint Cumulants of Bilinear Forms.- 2.4c Moments and Cumulants in the Singular Normal Case.- 2.4d Cumulants 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 QuadraticForm.- 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 theRepresentation Matrix over the Orthogonal Group.- Glossary of Symbols.- Author Index.