Advertisement

Distribution of a Quadratic Form in Normal Vectors

(Multivariate Non-Central Case)
  • C. G. Khatri
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 17)

Summary

Let the column vectors of \(\mathop {\text{X}}\limits_ \sim :\) pxn be distributed as independent normals with common covariance matrix \(\mathop \sum \limits_ \sim \). Then, the quadratic form in normal vectors is denoted by \(\mathop {\text{X}}\limits_ \sim \mathop {\text{A}}\limits_ \sim \mathop {\text{X}}\limits_ \sim '\, = \,\mathop {\text{S}}\limits_ \sim \) where \(\mathop {\text{A}}\limits_ \sim :\,{\text{nxn}}\) is a symmetric matrix which is assumed to be positive definite. This paper deals with a series representation of the density function of \(\mathop {\text{S}}\limits_ \sim \) when \({\text{E(}}\mathop {\text{X}}\limits_ \sim {\text{)}}\, \ne \,\mathop 0\limits_ \sim \), extending the idea of the author (1971) and Kotz et al. (1967b) to the multivariate non- central case. It is pointed out that this method gives a result which is not easy to obtain directly by integrating over an orthogonal space in the sense of James (1964).

Key Words

Quadratic form in normal vectors series representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Constantine, A. G. (1963). Ann. Math. Statist. 34, 12 70–1285.MathSciNetGoogle Scholar
  2. [2]
    Gurland, J. (1955). Ann. Math. Statist. 26, 122–127. Corrections in (1962), Ann. Math. Statist. 33, 813.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Hayakawa, T. (1966). Ann. Inst. Statist. Math. 18, 191–200.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    James, A. T. (1964). Ann. Math. Statist. 35, 475–501.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Khatri, C. G. (1966). Ann. Math. Statist. 37, 468–479.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Khatri, C. G. (1971). J. Multivariate Anal. 1 199–214.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Kotz, S., Johnson, N. L. and Boyd, D. W. (1967a). Ann. Math. Statist. 38, 823–837.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Kotz, S., Johnson, N. L. and Boyd, D. W. (1967b). Ann. Math. Statist. 38, 838–848.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Pachares, J. (1955). Ann. Math. Statist. 26, 128–131.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Robbins, H. (1948). Ann. Math. Statist. 19, 266–270.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Ruben, H. (1962). Ann. Math. Statist. 542–570.Google Scholar
  12. [12]
    Ruben, H. (1963). Ann. Math. Statist. 34, 1582–1584.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Shah, B. K. (1963). Ann. Math. Statist. 34, 186–190.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Shah, B. K. (1968). Ann. Math. Statist. 39, 18, 1090.Google Scholar
  15. [15]
    Shah, B. K. (1970). Ann. Math. Statist. 41, 692–697.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Shah, B. K. and Khatri, C. G. (1961). Ann. Math. Statist. 32, 883–887. Corrections in (1963), Ann. Math. Statist. 34, 673.MathSciNetCrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht-Holland 1975

Authors and Affiliations

  • C. G. Khatri
    • 1
  1. 1.Gujarat UniversityAhmedabadIndia

Personalised recommendations