Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data

Abstract

We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the families both theoretically (relevant properties and distinctive features) and with a Monte Carlo study (comparing the fitting abilities in finite samples).

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

“…Currently, the availability of larger data sets requires consideration of more flexible families of distributions and, at the same time, more powerful computing resources allow researchers to explore new avenues. An overview of recent developments in this domain can be obtained by the accounts of Jones [4] and contributions to the ensuing discussion, as well as that of Babić et al [5]; review in [4] focuses mainly on the univariate case, while [5] deals with the multivariate case.…”

“…More complex formulations, such as some of those discussed in the review paper [5], allow for a separate tail parameter for each component variable. This offers an increased level of flexibility, but at some cost in terms of mathematical tractability.…”

“…Classical specifications for U, V, where we now drop the subscript i for simplicity, are as follows: (35) where χ 1 denotes the square root of a χ 2 1 random variable, also called half-normal distribution. The distribution of R = V−U under specification (35) is of the SN type, taking into account its additive representation (5). Simple algebra shows that the SN parameters are related to those in (35) by ω 2 = σ 2 u + σ 2 v , α = −σ u /σ v .…”

TailCoR

“…While EVT-based literature is semi-parametric (in the sense that only the asymptotic tail is parametrized), another branch of the literature parametrises the whole distribution while assuming heavy tails -see Babić et al (2019) for a survey. The scale mixtures of multinormal distributions are particularly relevant since the elliptical and the normal mean-variance mixture distributions are used in this article.…”

Variance Ranking for Multi-Classed Imbalanced Datasets: A Case Study of One-Versus-All

“…Ref. [14] has applied the level of skewness to a multivariate dataset and implies that this relates to the distributions and ultimately the classes. Some researchers like [15,16] applied a deep learning algorithm to solve the imbalanced scenario of Malware detection.…”