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On the moments of ratios of quadratic forms in normal random variables

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  • Bao, Yong
  • Kan, Raymond

Abstract

In this paper, we present both integral and infinite series expressions of μqp≡E[(x′Ax)p/(x′Bx)q] when x∼N(μ,In), where p, q are nonnegative real numbers, A is a symmetric matrix, and B is a positive semi-definite matrix. We also present efficient numerical methods for computing μqp under each approach.

Suggested Citation

  • Bao, Yong & Kan, Raymond, 2013. "On the moments of ratios of quadratic forms in normal random variables," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 229-245.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:229-245
    DOI: 10.1016/j.jmva.2013.03.002
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    References listed on IDEAS

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    1. 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.
    2. Roberts, Leigh A., 1995. "On the Existence of Moments of Ratios of Quadratic Forms," Econometric Theory, Cambridge University Press, vol. 11(4), pages 750-774, August.
    3. Smith, Murray D., 1989. "On the expectation of a ratio of quadratic forms in normal variables," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 244-257, November.
    4. Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(2), pages 195-207, April.
    5. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
    6. Sawa, Takamitsu, 1972. "Finite-Sample Properties of the k-Class Estimators," Econometrica, Econometric Society, vol. 40(4), pages 653-680, July.
    7. 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.
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    Cited by:

    1. Li, Min & Liu, Min-Qian & Wang, Xiao-Lei & Zhou, Yong-Dao, 2020. "Prediction for computer experiments with both quantitative and qualitative factors," Statistics & Probability Letters, Elsevier, vol. 165(C).
    2. Yong Bao & Aman Ullah, 2021. "Analytical Finite Sample Econometrics: From A. L. Nagar to Now," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 17-37, December.

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