Extending the multivariate generalised t and generalised VG distributions

The GGH family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting GGH family encompasses the multivariate generalised hyperbolic (GH), which itself contains the multivariate t and multivariate Variance-Gamma (VG) 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 VG as special cases. Our approach unifies into a single GH-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 VG-type cases. The GGH 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 VG [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.