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

Von Mises-Fisher Elliptical Distribution

Modern probabilistic learning systems mainly assume symmetric distributions, however, real-world data typically obey skewed distributions and are thus not adequately modeled through symmetric distributions. To address this issue, a generalization of symmetric distributions called elliptical distributions are increasingly used, together with further improvements based on skewed elliptical distributions. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose a novel approach based on the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of skewed elliptical distributions. The analysis shows that this not only allows us to design and implement nonsymmetric learning systems but also provides a physically meaningful and intuitive way of generalizing skewed distributions. For rigor, the proposed framework is proven to share important and desirable properties with its symmetric counterpart. The proposed vMF distribution is demonstrated to be easy to generate and stable to estimate, both theoretically and through examples.

The Linear Skew-t Distribution and Its Properties

The aim of this expository paper is to present the properties of the linear skew-t distribution, which is a specific example of a symmetry modulated-distribution. The skewing function remains the distribution function of Student’s t, but its argument is simpler than that used for the standard skew-t. The linear skew-t offers different insights, for example, different moments and tail behavior, and can be simpler to use for empirical work. It is shown that the distribution may be expressed as a hidden truncation model. The paper describes an extended version of the distribution that is analogous to the extended skew-t. For certain parameter values, the distribution is bimodal. The paper presents expressions for the moments of the distribution and shows that numerical integration methods are required. A multivariate version of the distribution is described. The bivariate version of the distribution may also be bimodal. The distribution is not closed under marginalization, and stochastic ordering is not satisfied. The properties of the distribution are illustrated with numerous examples of the density functions, table of moments and critical values. The results in this paper suggest that the linear skew-t may be useful for some applications, but that it should be used with care for methodological work.

An overview of heavy-tail extensions of multivariate Gaussian distribution and their relations

Many extensions of the multivariate normal distribution to heavy-tailed distributions are proposed in the literature, which includes scale Gaussian mixture distribution, elliptical distribution, generalized elliptical distribution and transelliptical distribution. The inferences for each family of distributions are well studied. However, extensions are overlapped or similar to each other, and it is hard to differentiate one extension from the other. For this reason, in practice, researchers simply pick one of many extensions and apply it to the analysis. In this paper, to enlighten practitioners who should conduct statistical procedures not based on their preferences but based on how data look like, we comparatively review various extensions and their estimators. Also, we fully investigate the inclusion and exclusion relations of different extensions by Venn diagrams and examples. Moreover, in the numerical study, we illustrate visual differences of the extensions by bivariate plots and analyze different scatter matrix estimators based on the microarray data.

Information–Theoretic Aspects of Location Parameter Estimation under Skew–Normal Settings

In several applications, the assumption of normality is often violated in data with some level of skewness, so skewness affects the mean’s estimation. The class of skew–normal distributions is considered, given their flexibility for modeling data with asymmetry parameter. In this paper, we considered two location parameter (μ) estimation methods in the skew–normal setting, where the coefficient of variation and the skewness parameter are known. Specifically, the least square estimator (LSE) and the best unbiased estimator (BUE) for μ are considered. The properties for BUE (which dominates LSE) using classic theorems of information theory are explored, which provides a way to measure the uncertainty of location parameter estimations. Specifically, inequalities based on convexity property enable obtaining lower and upper bounds for differential entropy and Fisher information. Some simulations illustrate the behavior of differential entropy and Fisher information bounds.

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

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.

Bivariate Power-Skew-Elliptical Distribution

In this article, we introduce a power-skew-elliptical (PSE) distribution in the bivariate setting. The new bivariate model arises in the context of conditionally specified distributions. The proposed bivariate model is an absolutely continuous distribution whose marginals are univariate PSE distributions. The special case of the bivariate power-skew-normal (BPSN) distribution is studied in details. General properties of the BPSN distribution are derived and the estimation of the unknown parameters by maximum pseudo-likelihood is discussed. Further, a sandwich type matrix, which is a consistent estimator for the asymptotic covariance matrix of the maximum likelihood (ML) estimator is determined. Two applications for real data of the proposed bivariate distribution is provided for illustrative purposes.