Multivariate Calculation

1. Front Matter

   Pages i-xvi

   PDF

2. Introduction and Brief Survey

   • Roger H. Farrell

   Pages 1-12

3. Transforms

   • Roger H. Farrell

   Pages 13-29

4. Locally Compact Groups and Haar Measure

   • Roger H. Farrell

   Pages 30-50

5. Wishart’s Paper

   • Roger H. Farrell

   Pages 51-63

6. The Fubini-Type Theorems of Karlin

   • Roger H. Farrell

   Pages 64-86

7. Manifolds and Exterior Differential Forms

   • Roger H. Farrell

   Pages 87-109

8. Invariant Measures on Manifolds

   • Roger H. Farrell

   Pages 110-123

9. Matrices, Operators, Null Sets

   • Roger H. Farrell

   Pages 124-162

10. Examples Using Differential Forms

    • Roger H. Farrell

    Pages 163-177

11. Cross-Sections and Maximal Invariants

    • Roger H. Farrell

    Pages 178-195

12. Random Variable Techniques

    • Roger H. Farrell

    Pages 196-229

13. The Construction of Zonal Polynomials

    • Roger H. Farrell

    Pages 230-297

14. Problems for Users of Zonal Polynomials

    • Roger H. Farrell

    Pages 298-314

15. Multivariate Inequalities

    • Roger H. Farrell

    Pages 315-357

16. Back Matter

    Pages 358-376

    PDF

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books of the time and also much material about multivariate computations existed only in the journal literature or in unpublished sets oflecture notes.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.

• Covariance matrix
• Maxima
• Random variable
• Variance
• correlation

• Mathematics Department, Cornell University, Ithaca, USA

  Roger H. Farrell

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