Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis

Xueying Yu; Chuancun Yin; Xueying Yu; Chuancun Yin

doi:10.3934/math.2023370

AIMS Mathematics

2023, Volume 8, Issue 3: 7346-7376. doi: 10.3934/math.2023370

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Research article

Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis

Xueying Yu ,
Chuancun Yin ^,

School of Statistics and Data Science, Qufu Normal University, Qufu, Shandong 273165, China

Received: 03 October 2022 Revised: 31 December 2022 Accepted: 03 January 2023 Published: 13 January 2023

The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos ^[1], we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew-$ t $, skew-Pearson type Ⅶ and skew-Pearson type Ⅱ distributions as examples.
- Mardia's measures,
- moments,
- multivariate elliptical distributions,
- multivariate skew-elliptical distributions,
- Song's measure of kurtosis
Citation: Xueying Yu, Chuancun Yin. Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis[J]. AIMS Mathematics, 2023, 8(3): 7346-7376. doi: 10.3934/math.2023370

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Abstract

The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos ^[1], we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew-$ t $, skew-Pearson type Ⅶ and skew-Pearson type Ⅱ distributions as examples.

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