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Explicit expressions for joint moments of $n$-dimensional elliptical distributions

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  • Baishuai Zuo
  • Chuancun Yin
  • Narayanaswamy Balakrishnan

Abstract

Inspired by Stein's lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive $E[X_{1}^{2}f(\mathbf{X})]$ for any measurable function $f$ satisfying some regularity conditions. Then, by applying this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of $E[X_{1}^{2}f(\mathbf{X})]$ for multivariate Student-$t$, logistic and Laplace distributions.

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  • Baishuai Zuo & Chuancun Yin & Narayanaswamy Balakrishnan, 2020. "Explicit expressions for joint moments of $n$-dimensional elliptical distributions," Papers 2007.09349, arXiv.org, revised Aug 2020.
    • Handle: RePEc:arx:papers:2007.09349
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    References listed on IDEAS

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    1. Song, Iickho & Lee, Seungwon, 2015. "Explicit formulae for product moments of multivariate Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 27-34.
    2. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2018. "A multivariate tail covariance measure for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 27-35.
    3. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    4. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
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