Abstract
The Riesz distribution for real normed division algebras is derived in this work. Then two versions of these distributions are proposed and some of their properties are studied.
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Baez JC (2002) The octonions 39:145–205
Bhavsar CD (2002) Asymptotic distributions of likelihood ratio criteria for two testing problems. Kybernetes 29(4):510–517
Casalis M, Letac G (1996) The Lukascs–Olkin–Rubin characterization of Wishart distributions on symmetric cones. Ann Stat 24:768–786
Constantine AC (1963) Noncentral distribution problems in multivariate analysis. Ann Math Stat 34: 1270–1285
Davis AW (1980) Invariant polynomials with two matrix arguments, extending the zonal polynomials. In: Krishnaiah PR (ed) Multivariate analysis V. North-Holland Publishing Company, Amsterdam, pp 287–299
Dray T, Manogue CA (1999) The exceptional Jordan eigenvalue problem. Int J Theor Phys 38(11):2901–2916
Díaz-García JA (2011) Generalizations of some properties of invariant polynomials with matrix arguments. Appl Math (Warsaw) 38(4):469–475
Díaz-García JA, Gutiérrez-Jáimez R (2009) Special functions: integral properties of Jack polynomials, hypergeometric functions and Invariant polynomials. http://arxiv.org/abs/0909.1988. Also submited
Díaz-García JA, Gutiérrez-Jáimez R (2011) On Wishart distribution: some extensions. Linear Algebra Appl 435:1296–1310
Díaz-García JA, Gutiérrez-Jáimez R (2012a) Matricvariate and matrix multivariate T distributions and associated distributions. Metrika 75(7):963–976
Díaz-García JA, Gutiérrez-Jáimez R (2012b) An identity of Jack polynomials. J Iran. Stat. Soc. 11(1):87–92
Díaz-García JA, Gutiérrez-Jáimez R (2013) Spherical ensembles. Linear Algebra Appl 438:3174–3201
Faraut J, Korányi A (1994) Analysis on symmetric cones. Oxford mathematical monographs. Clarendon Press, Oxford
Forrester PJ (2010) Log-gases and random matrices. London Mathematical Society monographs. Princeton University Press, Princeton, NJ
Goodman NR (1963) Statistical analysis based on a certain multivariate complex Gaussian distribution (an introdiction). Ann Math Stati 34(1):152–177
Gross KI, Richards DSTP (1987) Special functions of matrix argument I: algebraic induction zonal polynomials and hypergeometric functions. Trans Am Math Soc 301(2):475–501
Hassairi A, Lajmi S (2001) Riesz exponential families on symmetric cones. J Theor Probab 14:927–948
Hassairi A, Lajmi A, Zine R (2005) Beta-Riesz distributions on symmetric cones. J Stat Plan Inf 133: 387–404
Herz CS (1955) Bessel functions of matrix argument. Ann Math 61(3):474–523
James AT (1961) Zonal polynomials of the real positive definite symmetric matrices. Ann Math 35:456–469
James AT (1964) Distribution of matrix variate and latent roots derived from normal samples. Ann Math Stat 35:475–501
Kabe DG (1984) Classical statistical analysis based on a certain hypercomplex multivariate normal distribution. Metrika 31:63–76
Khatri CG (1966) On certain distribution problems based on positive definite quadratic functions in normal vector. Ann Math Stat 37:468–479
Li F, Xue Y (2009) Zonal polynomials and hypergeometric functions of quaternion matrix argument. Commun Stat Theory Methods 38(8):1184–1206
Massam H (1994) An exact decomposition theorem and unified view of some related distributions for a class of exponential transformation models on symmetric cones. Ann Stat 22(1):369–394
Micheas AC, Dey DK, Mardia KV (2006) Complex elliptical distribution with application to shape theory. J Stat Plan Inference 136:2961–2982
Muirhead RJ (1982) Aspects of Multivariate Statistical Theory. Wiley, New York
Riesz M (1949) L’intégrale de Riemann–Liouville et le problème de Cauchy. Acta Math 81:1–223
Sawyer P (1997) Spherical functions on symmetric cones. Trans Am Math Soc 349:3569–3584
Wooding RA (1956) The multivariate distribution of complex normal variables. Biometrika 43(1):212–215
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The author wish to thank the Editor and the anonymous reviewers for their constructive comments on the preliminary version of this paper.
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Díaz-García, J.A. On Riesz distribution. Metrika 77, 469–481 (2014). https://doi.org/10.1007/s00184-013-0449-5
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DOI: https://doi.org/10.1007/s00184-013-0449-5