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Evaluating the density of ratios of noncentral quadratic forms in normal variables

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  • Broda, S.
  • Paolella, M.S.

Abstract

Two computable expressions for the exact density of a ratio of quadratic forms in Gaussian random vectors are derived, one of which is restricted to special cases of the problem. Ratios of this type are ubiquitous in econometrics, but their density, unlike the corresponding cumulative distribution function, has not received much attention to date. The new algorithms complement those available for the latter. The included performance study demonstrates the accuracy of the two algorithms, both absolute and relative to each other, and allows general recommendations on their use to be made.

Suggested Citation

  • Broda, S. & Paolella, M.S., 2009. "Evaluating the density of ratios of noncentral quadratic forms in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1264-1270, February.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:4:p:1264-1270
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    1. R. W. Farebrother, 1984. "The Distribution of a Linear Combination of Central X2 Random Variables a Remark on as 153: Pan's Procedure for the Tail Probabilities of the Durbin–Watson Statistic," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(3), pages 363-366, November.
    2. Broda, Simon & Carstensen, Kai & Paolella, Marc S., 2007. "Bias-adjusted estimation in the ARX(1) model," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3355-3367, April.
    3. Magnus, Jan R., 1985. "On Differentiating Eigenvalues and Eigenvectors," Econometric Theory, Cambridge University Press, vol. 1(2), pages 179-191, August.
    4. Paolella, Marc S., 2003. "Computing moments of ratios of quadratic forms in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 42(3), pages 313-331, March.
    5. Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.
    6. Marsh, Patrick W.N., 1998. "Saddlepoint Approximations For Noncentral Quadratic Forms," Econometric Theory, Cambridge University Press, vol. 14(5), pages 539-559, October.
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    Cited by:

    1. Pronzato, Luc, 2019. "Sensitivity analysis via Karhunen–Loève expansion of a random field model: Estimation of Sobol’ indices and experimental design," Reliability Engineering and System Safety, Elsevier, vol. 187(C), pages 93-109.

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