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The Exact Moments of a Ratio of Quadratic Forms in Normal Variables


  • Jan R. Magnus


The exact moments of x'Ax/x'Bx are obtained, where x is a normally distributed vector with some mean (possibly nonzero) and positive definite covariance matrix, A is symmetric and B positive semi definite. These moments appear as simple integrals which can be evaluated numerically in a straightforward manner. In addition, the precise conditions for the existence of the moments are found. Some related results are also reported.

Suggested Citation

  • Jan R. Magnus, 1986. "The Exact Moments of a Ratio of Quadratic Forms in Normal Variables," Annals of Economics and Statistics, GENES, issue 4, pages 95-109.
  • Handle: RePEc:adr:anecst:y:1986:i:4:p:95-109

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    Cited by:

    1. Long Qu & Tobias Guennel & Scott L. Marshall, 2013. "Linear Score Tests for Variance Components in Linear Mixed Models and Applications to Genetic Association Studies," Biometrics, The International Biometric Society, vol. 69(4), pages 883-892, December.
    2. Marcus J. Chambers, 2015. "The Calculation of Some Limiting Distributions Arising in Near-Integrated Models with GLS Detrending," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(4), pages 562-586, July.
    3. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 918, Economics Division, School of Social Sciences, University of Southampton.
    4. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.
    5. Antonis Demos & Dimitra Kyriakopoulou, 2018. "Finite Sample Theory and Bias Correction of Maximum Likelihood Estimators in the EGARCH Model," DEOS Working Papers 1802, Athens University of Economics and Business.

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