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Extending the multivariate generalised t and generalised VG distributions

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  • Fung, Thomas
  • Seneta, Eugene

Abstract

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.

Suggested Citation

  • Fung, Thomas & Seneta, Eugene, 2010. "Extending the multivariate generalised t and generalised VG distributions," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 154-164, January.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:1:p:154-164
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    References listed on IDEAS

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    1. Press, S. J., 1972. "Multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 2(4), pages 444-462, December.
    2. Harrar, Solomon W. & Seneta, Eugene & Gupta, Arjun K., 2006. "Duality between matrix variate t and matrix variate V.G. distributions," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1467-1475, July.
    3. Arslan, Olcay, 2005. "A new class of multivariate distributions: Scale mixture of Kotz-type distributions," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 18-28, November.
    4. Elisa Luciano & Wim Schoutens, 2006. "A multivariate jump-driven financial asset model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 385-402.
    5. Fung, Thomas & Seneta, Eugene, 2008. "A characterisation of scale mixtures of the uniform distribution," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2883-2888, December.
    6. Simon Hurst & Eckhard Platen & Svetlozar Rachev, 1997. "Subordinated Market Index Models: A Comparison," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 4(2), pages 97-124, May.
    7. McDonald, James B. & Newey, Whitney K., 1988. "Partially Adaptive Estimation of Regression Models via the Generalized T Distribution," Econometric Theory, Cambridge University Press, vol. 4(3), pages 428-457, December.
    8. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
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    1. Boris Buchmann & Benjamin Kaehler & Ross Maller & Alexander Szimayer, 2015. "Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing," Papers 1502.03901, arXiv.org, revised Oct 2016.
    2. Buchmann, Boris & Kaehler, Benjamin & Maller, Ross & Szimayer, Alexander, 2017. "Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2208-2242.
    3. Finlay, Richard & Seneta, Eugene, 2012. "A Generalized Hyperbolic model for a risky asset with dependence," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2164-2169.
    4. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 73-85.

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