Extending the multivariate generalised \(t\) and generalised \(VG\) distributions. (English) Zbl 1182.62121
The GGH family of multivariate distributions is obtained by scale mixing of the exponential power distribution using the extended generalized 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 generalized \(t\) distributions. J. Multivariate Anal. 89, No. 2, 329–337 (2004; Zbl 1049.62058)] and a new generalisation of the VG as special cases. The approach unifies into a single GH-type family the hitherto separately treated t-type [O. Arslan, A new class of multivariate distributions: scale mixture of Kotz-type distributions. Stat. Probab. Lett. 75, No. 1, 18–28 (2005; Zbl 1082.62047); Variance-mean mixture of Kotz-type distributions. Commun. Stat., Theory Methods 38, 272–284 (2009)] and VG-type cases. The GGH distribution is dual to the distribution obtained by analogously mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate VG [S.W. Harrar, E. Seneta and A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions. J. Multivariate Anal. 97, No. 6, 1467–1475 (2006; Zbl 1119.62048)] does however extend in some sense to their generalisations.
Reviewer: Jaroslaw Bartoszewicz (Wrocław)
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 33C90 | Applications of hypergeometric functions |
| 60E05 | Probability distributions: general theory |
Keywords:
duality; generalized hyperbolic distributions; generalized t distributions; exponential power distributions; variance-gamma distributionsReferences:
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