Abstract
The density of the quotient of two non-negative quadratic forms in normal variables is considered. The covariance matrix of these variables is arbitrary. The result is useful in the study of the robustness of theF-test with respect to errors of the first and second kind. An explicit expression for this density is given in the form of a proper Riemann-integral on a finite interval, suitable for numerical calculation.
Similar content being viewed by others
References
Cramer, H. (1963)Mathematical Methods of Statistics, Princeton University Press, Princeton.
Geary, R. C. (1944) Extension of a theorem by Harold Cramer.Journal of the Royal Statistical Society.17, 56–57.
Gradshteyn, I. S. and Ryzhik, I. M. (1965)Tables of Integrals, Series and Products, 4th edn, Academic Press, New York.
Lugannani, R. and Rice, S. O. (1984) Distribution of the ratio of quadratic forms in normal variables: numerical methods.SIAM Journal of Statistics and Computing,5, 476–488.
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.
Scheffé, H. (1959)The Analysis of Variance, Wiley, New York.
Seber, G. A. F. (1977)Linear Regression Analysis, Wiley, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van der Genugten, B.B. Density of the quotient of non-negative quadratic forms in normal variables with application to the F-statistic. Stat Comput 2, 179–182 (1992). https://doi.org/10.1007/BF01889677
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01889677