Extending the multivariate generalised tt and generalised VGVG distributions

The GGHGGH family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting GGHGGH family encompasses the multivariate generalised hyperbolic (GHGH), which itself contains the multivariate tt and multivariate Variance-Gamma (VGVG) distributions as special cases. It also contains the generalised multivariate tt distribution [O. Arslan, Family of multivariate generalised tt distribution, Journal of Multivariate Analysis 89 (2004) 329–337] and a new generalisation of the VGVG as special cases. Our approach unifies into a single GHGH-type family the hitherto separately treated tt-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 VGVG-type cases. The GGHGGH distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate tt and multivariate VGVG [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate tt and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467–1475] does however extend in one sense to their generalisations.