Multivariate Calculation

Like some of my colleagues, in my earlier years I found the multivariate Jacobian calculations horrible and unbelievable. As I listened and read during the years 1956 to 1974 I continually saw alternatives to the Jacobian and variable change method of computing probability density functions. Further, it was made clear by the work of A. T. James that computation of the density functions of the sets of roots of determinental equations required a method other than Jacobian calculations and that the densities could be calculated using differential forms on manifolds. It had become clear from the work ofC S. Herz and A. T. James that the expression of the noncentral multivariate density functions required integration with respect to Haar measures on locally compact groups. Material on manifolds and locally compact groups had not yet reached the pages of multivariate books of the time and also much material about multivariate computations existed only in the journal literature or in unpublished sets oflecture notes. In spirit, being more a mathematician than a statistician, the urge to write a book giving an integrated treatment of these topics found expression in 1974-1975 when I took a one year medical leave of absence from Cornell University. During this period I wrote Techniques of Multivariate Calculation. Writing a coherent treatment of the various methods made obvious re­ quired background material.

1 Introduction and Brief Survey.- 1.1. Aspects of Multivariate Analysis.- 1.2. On the Organization of the Book.- 1.3. Sources and the Literature.- 1.4. Notations.- 2 Transforms.- 2.0. Introduction.- 2.1. Definitions and Uniqueness.- 2.2. The Multivariate Normal Density Functions.- 2.3. Noncentral Chi-Square, F-, and t-Density Functions.- 2.4. Inversion of Transforms and Hermite Polynomials.- 2.5. Inversion of the Laplace and Mellin Transforms.- 2.6. Examples in the Literature.- 3 Locally Compact Groups and Haar Measure.- 3.0. Introduction.- 3.1. Basic Point Set Topology.- 3.2. Quotient Spaces.- 3.3. Haar Measure.- 3.4. Factorization of Measures.- 3.5. Modular Functions.- 3.6. Differential Forms of Invariant Measures on Matrix Groups.- 3.7. Cross-Sections.- 3.8. Solvability, Amenability.- 4 Wishart’s Paper.- 4.0. Introduction.- 4.1. Wishart’s Argument.- 4.2. The Noncentral Wishart Density Function.- 4.3. James on Series, Rank 3.- 4.4. Related Problems.- 5 The Fubini-Type Theorems of Karlin.- 5.0. Introduction.- 5.1. The Noncentral t-Density.- 5.2. The Wishart Density Function.- 5.3. The Eigenvalues of the Covariance Matrix.- 5.4. The Generalized T.- 5.5. Remarks on Noncentral Problems.- 5.6. The Conditional Covariance Matrix.- 5.7. The Invariant S22-1/2S21S11-1S12S22-1/2.- 5.8. Some Problems.- 6 Manifolds and Exterior Differential Forms.- 6.0. Introduction.- 6.1. Basic Structural Definitions and Assumptions.- 6.2. Multilinear Forms, Algebraic Theory.- 6.3. Differential Forms and the Operator d.- 6.4. Theory of Integration.- 6.5. Transformation of Manifolds.- 6.6. Lemmas on Multiplicative Functionals.- 6.7. Problems.- 7 Invariant Measures on Manifolds.- 7.0. Introduction.- 7.1. ?nh.- 7.2. Lower Triangular Matrices, Left and Right Multiplication.- 7.3. S(h).- 7.4. The Orthogonal Group O(n).- 7.5. Grassman Manifolds Gk,n-k.- 7.6. Stiefel Manifolds Vk,n.- 7.7. Total Mass on the Stiefel Manifold, k = 1.- 7.8. Mass on the Stiefel Manifold, General Case.- 7.9. Total Mass on the Grassman Manifold Gk,n-k.- 7.10. Problems.- 8 Matrices, Operators, Null Sets.- 8.0. Introduction.- 8.1. Matrix Decompositions.- 8.2. Canonical Correlations.- 8.3. Operators and Gaussian Processes.- 8.4. Sets of Zero Measure.- 8.5. Problems.- 9 Examples Using Differential Forms.- 9.0. Introduction.- 9.1. Density Function of the Critical Angles.- 9.2. Hotelling T2.- 9.3. Eigenvalues of the Sample Covariance Matrix XtX.- 9.4. Problems.- 10 Cross-Sections and Maximal Invariants.- 10.0. Introduction.- 10.1. Basic Theory.- 10.2. Examples.- 10.3. Examples: The Noncentral Multivariate Beta Density Function.- 10.4. Modifications of the Basic Theory.- 10.5. Problems.- 11 Random Variable Techniques.- 11.0. Introduction.- 11.1. Random Orthogonal Matrices.- 11.2. Decomposition of the Sample Covariance Matrix Using Random Variable Techniques. The Bartlett Decomposition.- 11.3. The Generalized Variance, Zero Means.- 11.4. Noneentral Wishart, Rank One Means.- 11.5. Hotelling T2 Statistic, Noneentral Case.- 11.6. Generalized Variance, Nonzero Means.- 11.7. Distribution of the Sample Correlation Coefficient.- 11.8. Multiple Correlation, Algebraic Manipulations.- 11.9. Distribution of the Multiple Correlation Coefficient.- 11.10. BLUE: Best Linear Unbiased Estimation, an Algebraic Theory.- 11.11. The Gauss―Markov Equations and Their Solution.- 11.12. Normal Theory. Idempotents and Chi-Squares.- 11.13. Problems.- 12 The Construction of Zonal Polynomials.- 12.0. Introduction.- 12.1. Kronecker Products and Homogeneous Polynomials.- 12.2. Symmetric Polynomials in n Variables.- 12.3. The Symmetric Group Algebra.- 12.4. Young’s Symmetrizers.- 12.5. Realization of the Group Algebra as Linear Transformations.- 12.6. The Center of the Bi-Symmetric Matrices, as an Algebra.- 12.7. Homogeneous Polynomials II. Two-Sided Unitary Invariance.- 12.8. Diagonal Matrices.- 12.9. Polynomials of Diagonal Matrices X.- 12.10. Zonal Polynomials of Real Matrices.- 12.11. Alternative Definitions of Zonal Polynomials. Group Characters.- 12.12. Third Construction of Zonal Polynomials. The Converse Theorem.- 12.13. Zonal Polynomials as Eigenfunctions. Takemura’s Idea.- 12.14. The Integral Formula of Kates.- 13 Problems for Users of Zonal Polynomials.- 13.0. Introduction.- 13.1. Theory.- 13.2. Numerical Identities.- 13.3. Coefficients of Series.- 13.4. On Group Representations.- 13.5. First Construction of Zonal Polynomials.- 13.6. A Teaching Version.- 14 Multivariate Inequalities.- 14.0. Introduction.- 14.1. Lattice Ordering of the Positive Definite Matrices.- 14.2. Majorization.- 14.3. Eigenvalues and Singular Values.- 14.4. Results Related to Optimality Considerations.- 14.5. Loewner Ordering.- 14.6. Concave and Convex Measures.- 14.7. The FKG-Inequality.- 14.8. Problems.