Geometric scale mixtures of normal distributions

Recently, Kundu (2017) proposed a multivariate skewed distribution, termed the Geometric-Normal (GN) distribution, by compounding the multivariate normal distribution with the geometric distribution. This distribution is a viable alternative to Azzalini’s multivariate skew-normal distribution and possesses several desirable properties. This paper introduces a novel class of asymmetric distributions by compounding the geometric distribution with scale mixtures of normal distributions. This class constitutes a special case of the continuous mixtures of multivariate normal distributions introduced by Arellano-Valle and Azzalini (2021). The proposed multivariate distributions exhibit high flexibility, featuring heavy tails, multi-modality, and the ability to model skewness. We have also derived several properties of this class and discussed specific examples to illustrate its applications. The expectation–maximization algorithm was employed to calculate the maximum likelihood estimates of the unknown parameters. Simulation experiments have been performed to show the effectiveness of the proposed algorithm. For illustrative purposes, we have provided one multivariate data set where it has been observed that there exist members in the proposed class of models that can provide better fit compared to skew-normal, skew-t, and generalized hyperbolic distribution. In another example, it was demonstrated that when data generated from a heavy-tailed skew-t distribution is contaminated with noise, the proposed distributions offer a better fit compared to the skew-t distribution.