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Multivariate Skewness and Kurtosis

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Multivariate Statistical Methods

Part of the book series: Frontiers in Probability and the Statistical Sciences ((FROPROSTAS))

Abstract

A unified treatment of all currently available cumulant-based indices of multivariate skewness and kurtosis is provided in this chapter. They are expressed in terms of the third and fourth-order cumulant vectors, respectively. Such a treatment helps to reveal many subtle features and inter-connections among the existing indices. Computational formulae for obtaining these measures are provided for general families of multivariate distributions, yielding several new results and a systematic exposition of many known results. Based on this analysis, new measures of skewness and kurtosis are proposed.

For a given multivariate distribution, explicit formulae are provided for the asymptotic covariances of estimated cumulant vectors of the third and the fourth order, which are needed for showing the asymptotic normality of test statistics based on them. This allows us to extend several known results and provides ready-to-use expressions in terms of population cumulants and commutator matrices, for any symmetric and asymmetric distribution as long as the required moments exist.

Statistical inference for both multivariate skewness and kurtosis is studied.

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Authors and Affiliations

  1. Faculty of Informatics, Dept. of IT, University of Debrecen, Debrecen, Hungary

    György Terdik

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  1. György Terdik
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Terdik, G. (2021). Multivariate Skewness and Kurtosis. In: Multivariate Statistical Methods . Frontiers in Probability and the Statistical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-81392-5_6

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