Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T04:59:47.321Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Existence of Moments of Ratios of Quadratic Forms

Published online by Cambridge University Press:  11 February 2009

Leigh A. Roberts
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain simple and generally applicable conditions for the existence of mixed moments E([X′ AX]″/[X′BX]U) of the ratio of quadratic forms T = X′ AX/X′BX where A and B are n × n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.

If u ≤ v, the precise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T; if u > v, a prerequisite for existence of the (u, v)th mixed moment is the existence of the 2(u − v)th moment of X When Xis not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2v.

Type
Research Article
Information
Econometric Theory , Volume 11 , Issue 4 , August 1995 , pp. 750 - 774
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Cramer, H. (1946) Mathematical Methods of Statistics. Princeton, New Jersey: Princeton University Press.Google Scholar
Davis, A.W. (1976) Statistical distributions in univariate and multivariate Edgeworth populations. Biometrika 63, 661–670.CrossRefGoogle Scholar
Fang, K.-T., Kotz, S., & Ng, K.W. (1990) Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability. New York: Chapman and Hall.10.1007/978-1-4899-2937-2CrossRefGoogle Scholar
Gurland, J. (1953) Distribution of quadratic forms and ratios of quadratic forms. Annals of Mathematical Statistics 24, 416–427.10.1214/aoms/1177728981Google Scholar
Johnson, N.L. & Kotz, S. (1972) Distributions in Statistics: Continuous Univariate Distributions. Boston: Houghton Mifflin.Google Scholar
Jones, M.C. (1987) On moments of ratios of quadratic forms in normal variables. Statistics & Probability Letters 6, 129–136.10.1016/0167-7152(87)90086-1CrossRefGoogle Scholar
Lye, J.N. (1991) The numerical evaluation of the distribution function of a bilinear form to a quadratic form with econometric examples. Econometric Reviews 10, 217–234.10.1080/07474939108800205CrossRefGoogle Scholar
Magnus, J.R. (1986) The exact moments of a ratio of quadratic forms in normal variables. Annales d'Economie et de Statistique 4, 95–109.10.2307/20075629CrossRefGoogle Scholar
Mathai, A.M. & Provost, S.B. (1992) Quadratic Forms in Random Variables. Statistics: Text-books and Monographs, No. 126. New York: Marcel Dekker.Google Scholar
Moran, P.A.P. (1968) An Introduction to Probability Theory. Oxford: Clarendon Press.Google Scholar
Muirhead, R.J. (1982) Aspects of Multivariate Statistical Theory. New York: Wiley.CrossRefGoogle Scholar
Roberts, L.A. (1992) On ratios of random variables and generalised mortality rates Journal of Applied Probability 29, 268–279.10.2307/3214565CrossRefGoogle Scholar
Simmons, G.F. (1963) Introduction to Topology and Modern Analysis. New York: McGraw-Hill.Google Scholar
Smith, M.D. (1989) On the expectation of a ratio of quadratic forms in normal variables. Journal of Multivariate Analysis 31, 244–257.CrossRefGoogle Scholar
Sutradhar, B.C. & Bartlett, R.F. (1989) An approximation to the distribution of the ratio of two general quadratic forms with application to time series valued designs. Communications in Statistics, Part A: Theory and Methodology 18, 1563–1588.10.1080/03610928908829986CrossRefGoogle Scholar