A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications

Abstract

Within the context of flexible parametric families of distributions, much work has been dedicated in recent years to the theme of skew-symmetric distributions, or symmetry-modulated distributions, as we prefer to call them. The present contribution constitutes a review of this area, with special emphasis on multivariate skew-elliptical families, which represent the subset with more immediate impact on applications. After providing background information of the distribution theory aspects, we focus on the aspects more relevant for applied work. The exposition is targeted to non-specialists in this domain, although some general knowledge of probability and multivariate statistics is assumed. Given this aim, the mathematical profile is kept to the minimum required.

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

“…Its moments were derived in [47]. For further discussion on these distributions and their applications, see [32][33][34][35].…”

“…We run a Monte Carlo simulation where 1000 samples of size n = 1000 are generated from the St 3 distribution defined above. For each of the 1000 samples, measures of location, variance, skewness and kurtosis are computed applying Formulae (34) and (35) and the results of Lemmas 3 and 4. Empirical averages of the statistics are calculated, denoting the empirical expected value as E.…”

“…[32] provides some specific analytical properties of skew-symmetric distributions. Further discussion on these distributions and their applications can be found in [33][34][35].…”

Von Mises-Fisher Elliptical Distribution

“…The form of (3) represents a typical skewed elliptical distribution, of which Y 0 has also been extended to higher dimensions [17], [18], [19]. Importantly, the form of ( 3) is invariant under quadratic forms [9], and is closed under marginalisation and affine transforms; we refer to [20] for more detail. However, the estimation of the above skewed elliptical distributions can be ill-posed, especially regarding its shape (skewness) parameter.…”

On mean and/or variance mixtures of normal distributions

“…The quest for more flexible distributions has led to an ever growing development in the literature of parametric distribution. In the past two decades or so, intense interest has been in the area of skew or asymmetric distributions; see, for example, the book edited by Genton (2004), the monograph by Azzalini and Capitanio (2014), and the papers by Azzalini (2005), Arellano-Valle and Azzalini (2006) and Adcock and Azzalini (2020) for recent accounts of the literature on skew distributions. Many of these formulations belong to the class of skew-symmetric distributions, which is a generalization of the classical skew normal (SN) distribution by Azzalini and Dalla Valle (1996).…”