Abstract
The paper is devoted to relations between the matrix GIG and Wishart distributions. Our basic tool in the first part is a version of the Matsumoto-Yor property for matrix variables. This approach covers the following issues: the Herz identity for the Bessel function of matrix variate argument, characterization of a class of Wishart matrices and linear transformations of the matrix GIG distribution. The Bayesian Wishart model, studied in the second part, gives an alternative definition of the matrix GIG distribution. Such a model is characterized by linearity of conditional expectations and matrix GIG conditional distribution. It is also extended to Bayesian matrix GIG models, in the framework of which an interesting independence property is proved.
Similar content being viewed by others
References
Arnold BC, Castillo E and Sarabia J-M (1999). Conditional specification of statistical models. Springer, New York
Barndorff-Nielsen O, Blæsild P, Jensen JL and Jørgensen B (1982). Exponential transformation models. Proc R Soc Lond A 379: 41–65
Bernadac E (1995). Random continued fractions and inverse gaussian distribution on a symmetric cone. J Theor Probab 8: 221–259
Butler RW (1998). Generalized inverse Gaussian distributions and their Wishart connections. Scand J Stat 25: 69–75
Casalis M and Letac G (1994). Characterization of the Jorgensen set in generalized linear models. Test 3: 145–162
Diaconis P and Ylvisaker D (1979). Conjugate priors for exponential families. Ann Stat 7: 269–281
Geiger D and Heckerman D (1998). A characterization of the bivariate Wishart distribution. Probab Math Stat 18: 119–131
Geiger D and Heckerman D (2002). Parameter priors for directed acyclic graphical models and the characterization of several probability distributions. Ann Stat 30: 1412–1440
Gindikin SG (1975). Invariant generalized functions in homogeneous domains. Funct Anal Appl 9: 50–52
Herz CS (1955). Bessel functions of matrix argument. Ann Math 61: 474–523
Letac G (2003) Convergence de l’intégrale GIG, unpublished note, pp 1–2
Letac G, Massam H (2001) The normal quasi-Wishart distribution. In: Viana MAG, Richards DStP (eds) Algebraic methods in statistics and probability. AMS Contemp Math 287:231–239
Letac G and Seshadri V (1983). A characterization of the generalized inverse Gaussian distribution by continued fractions. Z Wahrsch Verw Geb 62: 482–489
Letac G and Wesołowski J (2000). An independence property for the product of GIG and gamma laws. Ann Probab 28: 1371–1383
Lukacs E (1955). A characterization of the gamma distribution. Ann Math Stat 26: 319–324
Massam H and Wesołowski J (2006). The Matsumoto-Yor property and the structure of the Wishart distribution. J Multivar Anal 97: 103–123
Muirhead RJ (1982). Aspects of multivariate statistical theory. Wiley, New York
Seshadri V (2003). Some properties of the matrix generalized inverse Gaussian distribution. In: Balakrishnan, N, Kannan, N and Srinivasan, MR (eds) Statistical methods and practice. Recent advances, pp 47–56. Narosa Publishing House, New Delhi
Vallois P (1989). Sur le passage de certaines marches aléatoires planes au-dessus d’une hyperbole équilatère. Ann Inst H Poincaré 25: 443–456
Wesołowski J (2002). The Matsumoto-Yor independence property for GIG and Gamma laws, revisited. Math Proc Camb Philos Soc 133: 153–161
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Seshadri, V., Wesołowski, J. More on connections between Wishart and matrix GIG distributions. Metrika 68, 219–232 (2008). https://doi.org/10.1007/s00184-007-0154-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-007-0154-3