On Wishart and noncentral Wishart distributions on symmetric cones
HTML articles powered by AMS MathViewer
- by Eberhard Mayerhofer PDF
- Trans. Amer. Math. Soc. 371 (2019), 7093-7109 Request permission
Abstract:
Necessary conditions for the existence of noncentral Wishart distributions are given. Our method relies on positivity properties of spherical polynomials on Euclidean Jordan algebras and advances an approach by Peddada and Richards [Ann. Probab. 19 (1991), pp. 868–874] where only a special case (positive semidefinite matrices, rank $1$ noncentrality parameter) is treated. Not only do the shape parameters need to be in the Wallach set—as is the case for Riesz measures—but also the rank of the noncentrality parameter is constrained by the size of the shape parameter. This rank condition has recently been proved with different methods for the special case of symmetric, positive semidefinite matrices (Letac and Massam; Graczyk, Malecki, and Mayerhofer).References
- M. Casalis and G. Letac, The Lukacs-Olkin-Rubin characterization of Wishart distributions on symmetric cones, Ann. Statist. 24 (1996), no. 2, 763–786. MR 1394987, DOI 10.1214/aos/1032894464
- M. Casalis and G. Letac, Characterization of the Jørgensen set in generalized linear models, Test 3 (1994), no. 1, 145–162. MR 1293112, DOI 10.1007/BF02562678
- Christa Cuchiero, Damir Filipović, Eberhard Mayerhofer, and Josef Teichmann, Affine processes on positive semidefinite matrices, Ann. Appl. Probab. 21 (2011), no. 2, 397–463. MR 2807963, DOI 10.1214/10-AAP710
- Christa Cuchiero, Martin Keller-Ressel, Eberhard Mayerhofer, and Josef Teichmann, Affine processes on symmetric cones, J. Theoret. Probab. 29 (2016), no. 2, 359–422. MR 3500404, DOI 10.1007/s10959-014-0580-x
- J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
- Ioana Dumitriu, Alan Edelman, and Gene Shuman, MOPS: multivariate orthogonal polynomials (symbolically), J. Symbolic Comput. 42 (2007), no. 6, 587–620. MR 2325917, DOI 10.1016/j.jsc.2007.01.005
- Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
- S. G. Gindikin, Invariant generalized functions in homogeneous domains, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 56–58 (Russian). MR 0377423
- Piotr Graczyk, Jacek Małecki, and Eberhard Mayerhofer, A characterization of Wishart processes and Wishart distributions, Stochastic Process. Appl. 128 (2018), no. 4, 1386–1404. MR 3769666, DOI 10.1016/j.spa.2017.07.010
- Jyoichi Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086–1110. MR 1226865, DOI 10.1137/0524064
- Gérard Letac and Hélène Massam, The Laplace transform $(\det s)^{-p}\exp \textrm {tr}(s^{-1}w)$ and the existence of non-central Wishart distributions, J. Multivariate Anal. 163 (2018), 96–110. MR 3732343, DOI 10.1016/j.jmva.2017.10.005
- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
- Hélène Massam and Erhard Neher, On transformations and determinants of Wishart variables on symmetric cones, J. Theoret. Probab. 10 (1997), no. 4, 867–902. MR 1481652, DOI 10.1023/A:1022658415699
- Robb J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. MR 652932
- Shyamal Das Peddada and Donald St. P. Richards, Proof of a conjecture of M. L. Eaton on the characteristic function of the Wishart distribution, Ann. Probab. 19 (1991), no. 2, 868–874. MR 1106290
- Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073, DOI 10.1016/0001-8708(89)90015-7
- John Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A (1928), 32–52.
Additional Information
- Eberhard Mayerhofer
- Affiliation: Department of Mathematics and Statistics, University of Limerick, Castletroy, Ireland
- MR Author ID: 750491
- Email: eberhard.mayerhofer@ul.ie
- Received by editor(s): October 23, 2017
- Received by editor(s) in revised form: February 20, 2018
- Published electronically: January 16, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7093-7109
- MSC (2010): Primary 60B15; Secondary 05A10, 05A05, 33C80
- DOI: https://doi.org/10.1090/tran/7754
- MathSciNet review: 3939571