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

Abstract

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.

Keywords: Elliptical distributions; generalized elliptical distributions; mutual relations; non-paranormal distributions; scatter matrix; transeliptical distributions.