Extending the multivariate generalised [formula omitted] and generalised [formula omitted] distributions

Abstract

The G G H family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting G G H family encompasses the multivariate generalised hyperbolic ( G H ), which itself contains the multivariate t and multivariate Variance-Gamma ( V G ) distributions as special cases. It also contains the generalised multivariate t distribution [O. Arslan, Family of multivariate generalised t distribution, Journal of Multivariate Analysis 89 (2004) 329–337] and a new generalisation of the V G as special cases. Our approach unifies into a single G H -type family the hitherto separately treated t -type [O. Arslan, A new class of multivariate distribution: Scale mixture of Kotz-type distributions, Statistics and Probability Letters 75 (2005) 18–28; O. Arslan, Variance–mean mixture of Kotz-type distributions, Communications in Statistics-Theory and Methods 38 (2009) 272–284] and V G -type cases. The G G H distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate V G [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467–1475] does however extend in one sense to their generalisations.