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New Methods for Multivariate Normal Moments

Author

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  • Christopher Stroude Withers

    (Callaghan Innovation (Formerly Industrial Research Ltd.), 101 Allington Road, Wellington 6012, New Zealand)

Abstract

Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4.

Suggested Citation

  • Christopher Stroude Withers, 2025. "New Methods for Multivariate Normal Moments," Stats, MDPI, vol. 8(2), pages 1-21, June.
    • Handle: RePEc:gam:jstats:v:8:y:2025:i:2:p:46-:d:1672622
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    References listed on IDEAS

    as
    1. Withers, Christopher S. & Nadarajah, Saralees, 2012. "Moments and cumulants for the complex Wishart," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 242-247.
    2. Phillips, Kem, 2010. "R Functions to Symbolically Compute the Central Moments of the Multivariate Normal Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(c01).
    3. C. S. Withers, 2024. "5th-Order Multivariate Edgeworth Expansions for Parametric Estimates," Mathematics, MDPI, vol. 12(6), pages 1-28, March.
    4. Withers, Christopher S. & Nadarajah, Saralees, 2014. "The dual multivariate Charlier and Edgeworth expansions," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 76-85.
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