Modelling directional dispersion through hyperspherical log‐splines

Summary.  We introduce the directionally dispersed class of multivariate distributions, a generalization of the elliptical class. By allowing dispersion of multivariate random variables to vary with direction it is possible to generate a very wide and flexible class of distributions. Directionally dispersed distributions have a simple form for their density, which extends a spherically symmetric density function by including a function D modelling directional dispersion. Under a mild condition, the class of distributions is shown to preserve both unimodality and moment existence. By adequately defining D , it is possible to generate skewed distributions. Using spline models on hyperspheres, we suggest a very flexible, yet practical, implementation for modelling directional dispersion in any dimension. Finally, we use the new class of distributions in a Bayesian regression set‐up and analyse the distributions of a set of biomedical measurements and a sample of US manufacturing firms.