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Multivariate Skew Distributions

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Multivariate Statistical Methods

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

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Abstract

In this chapter we provide a systematic treatment of several multivariate skew distributions. General formulae for cumulant vectors at least up to the fourth order are given, which are necessary for deriving the corresponding skewness and kurtosis measures discussed later in Sects. 6.1, 6.4, and 6.5.

In the first section all cumulants of multivariate skew-normal distributions and canonical fundamental skew-normal distributions are given. The Inverse Mill’s ratio and the central folded normal distribution are also discussed.

Section 5.1 is devoted to skew-spherical distributions. We start with two important classes of symmetric distributions, namely the spherically symmetric distributions and elliptically symmetric distributions, deriving in turn all their cumulants. Moment and cumulant parameters for spherical distributions are introduced, as they play an important role in the study of cumulants of multivariate elliptically contoured distributions. A canonical fundamental skew-spherical distribution is defined in a similar way to the canonical fundamental skew-normal distribution, and the first four cumulants are provided.

Cumulants for multivariate skew-t, scale mixtures of skew normal, multivariate skew-normal-Cauchy, and multivariate Laplace distributions are given in Sects. 5.4–5.6. The method in each case uses T -derivatives and T-cumulants.

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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 Skew Distributions. In: Multivariate Statistical Methods . Frontiers in Probability and the Statistical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-81392-5_5

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