Author | Mathai, A. M. author |
---|---|
Title | Bilinear Forms and Zonal Polynomials [electronic resource] / by A. M. Mathai, Serge B. Provost, Takesi Hayakawa |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4242-0 |
Descript | XII, 376 p. online resource |
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 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 Deยฃnitions 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 Cochranโs 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 XAXโ -- 5.5 Generalized Hotellingโs T02 -- 5.6 Andersonโs 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