Density of the quotient of non-negative quadratic forms in normal variables with application to the F-statistic
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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.
KeywordsRatio of quotient of quadratic forms in normal variables F-test,F-statistic numerical evaluation of probability densities
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- Cramer, H. (1963)Mathematical Methods of Statistics, Princeton University Press, Princeton.Google Scholar
- Geary, R. C. (1944) Extension of a theorem by Harold Cramer.Journal of the Royal Statistical Society.17, 56–57.Google Scholar
- Gradshteyn, I. S. and Ryzhik, I. M. (1965)Tables of Integrals, Series and Products, 4th edn, Academic Press, New York.Google Scholar
- 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.Google 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.Google Scholar
- Scheffé, H. (1959)The Analysis of Variance, Wiley, New York.Google Scholar
- Seber, G. A. F. (1977)Linear Regression Analysis, Wiley, New York.Google Scholar