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From moments of sum to moments of product


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  • Kan, Raymond
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    We provide an identity that relates the moment of a product of random variables to the moments of different linear combinations of the random variables. Applying this identity, we obtain new formulae for the expectation of the product of normally distributed random variables and the product of quadratic forms in normally distributed random variables. In addition, we generalize the formulae to the case of multivariate elliptically distributed random variables. Unlike existing formulae in the literature, our new formulae are extremely efficient for computational purposes.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 99 (2008)
    Issue (Month): 3 (March)
    Pages: 542-554

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    Handle: RePEc:eee:jmvana:v:99:y:2008:i:3:p:542-554

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    Keywords: Moments of product Product of random variables Product of quadratic forms;


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    1. Schott, James R., 2003. "Kronecker product permutation matrices and their application to moment matrices of the normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 177-190, October.
    2. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153219, Tilburg University.
    3. Berkane, Maia & Bentler, P. M., 1986. "Moments of elliptically distributed random variates," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 333-335, October.
    4. Blacher, René, 2003. "Multivariate quadratic forms of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 2-23, October.
    5. Magnus, J.R., 1978. "The moments of products of quadratic forms in normal variables," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153205, Tilburg University.
    6. Magnus, J.R., 1979. "The expectation of products of quadratic forms in normal variables: The practice Statistica Neerlandica," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153208, Tilburg University.
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    Cited by:
    1. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," CeMMAP working papers CWP14/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Vignat, C., 2012. "A generalized Isserlis theorem for location mixtures of Gaussian random vectors," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 67-71.
    3. Kan, Raymond & Wang, Xiaolu, 2010. "On the distribution of the sample autocorrelation coefficients," Journal of Econometrics, Elsevier, vol. 154(2), pages 101-121, February.


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