Flexible asymmetric multivariate distributions based on two-piece univariate distributions

Abstract

Classical symmetric distributions like the Gaussian are widely used. However, in reality data often display a lack of symmetry. Multiple distributions have been developed to specifically cope with asymmetric data. These can be grouped under the name "skewed distributions". In this paper we present a broad family of flexible multivariate skewed distributions for which statistical inference is a feasible task. The studied family of multivariate skewed distributions is derived by taking affine combinations of independent univariate distributions. These univariate distributions are members of a flexible family of asymmetric distributions and are an important basis for achieving statistical inference. Besides basic properties of the proposed distributions, also statistical inference based on a maximum likelihood approach is presented. We show that under some mild conditions, weak consistency and asymptotic normality of the maximum likelihood estimators hold. These results are backed up by a simulation study which confirms the developed theoretical results and some data examples to illustrate practical applicability.