Abstract
This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.
Highlights
This paper provides a systematic treatment and derivation of moments and cumulants of any order for spherically as well as elliptically symmetric multivariate distributions
We present an approach based on a vectorization of cumulants which leads to a natural and intuitive way to obtain multivariate moments and cumulants of any order; we make the point that this provides the simplest way to deal with this issue
The primary contributions of this paper may be summarized as: (a) providing formulae, valid for any order, for vectorized cumulants and moments of multivariate spherical and elliptically symmetric distributions; matrix structures can be readily obtained from these expressions; some examples are provided in Section 4; (b) introduce the so-called marginal moment and cumulant parameters for multivariate spherical and elliptically symmetric distributions—providing an extension of the results discussed in [6]; (c) provide formulae for all-order moments and cumulants for multivariate t and skew-symmetric t-distributions; results of this type, as far as we know, are not available in the literature
Summary
This paper provides a systematic treatment and derivation of moments and cumulants of any order for spherically as well as elliptically symmetric multivariate distributions. The primary contributions of this paper may be summarized as: (a) providing formulae, valid for any order, for vectorized cumulants and moments of multivariate spherical and elliptically symmetric distributions; matrix structures can be readily obtained from these expressions; some examples are provided in Section 4; (b) introduce the so-called marginal moment and cumulant parameters for multivariate spherical and elliptically symmetric distributions—providing an extension of the results discussed in [6]; (c) provide formulae for all-order moments and cumulants for multivariate t and skew-symmetric t-distributions; results of this type, as far as we know, are not available in the literature. The proofs are collected in a separate section at the end, viz
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