JTAS Ferreira and MFJ Steel
Modelling Directional Dispersion Through Hyperspherical Log-Splines
Abstract: We introduce the directionally dispersed class of multivariate distributions, a generalisation of the elliptical class. By allowing dispersion of multivariate random variables to vary with direction it is possible to generate a very wide and °exible 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 de¯ning D, it is possible to generate skewed distributions. Using spline models on hyperspheres, we suggest a very °exible, yet practical, implementation for modelling directional dispersion in any dimension. Finally, we use the new class of distributions in a Bayesian regression setup and analyse the distributions of a set of biomedical measurements and a sample of U.S. manufacturing ¯rms.
Keywords: Bayesian regression model, directional dispersion, elliptical distributions, existence of moments, modality, skewed distributions.