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On Mardia's and Song's measures of kurtosis in elliptical distributions

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  • Zografos, K.

Abstract

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure [beta]2,p and Song's measure . In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Renyi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.

Suggested Citation

  • Zografos, K., 2008. "On Mardia's and Song's measures of kurtosis in elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 858-879, May.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:5:p:858-879
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    References listed on IDEAS

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    1. Zografos, K. & Nadarajah, S., 2005. "Expressions for Rényi and Shannon entropies for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 71(1), pages 71-84, January.
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    Cited by:

    1. Loperfido, Nicola, 2020. "Some remarks on Koziol’s kurtosis," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    2. Leonenko, Nikolaj & Seleznjev, Oleg, 2010. "Statistical inference for the [epsilon]-entropy and the quadratic Rényi entropy," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1981-1994, October.
    3. Iwashita, Toshiya & Klar, Bernhard, 2014. "The joint distribution of Studentized residuals under elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 203-209.
    4. Baishuai Zuo & Narayanaswamy Balakrishnan & Chuancun Yin, 2023. "An analysis of multivariate measures of skewness and kurtosis of skew-elliptical distributions," Papers 2311.18176, arXiv.org.
    5. Batsidis, Apostolos & Zografos, Konstantinos, 2013. "A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 91-105.
    6. Faliva, Mario & Quatto, Piero & Zoia, Maria Grazia, 2018. "Gram–Charlier-like expansions of power-raised hyperbolic secant laws," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 229-234.
    7. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.

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