Wishart distributions on homogeneous cones. (English) Zbl 1058.62044
Summary: The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known families of Wishart distributions as special cases. The relationships to graphical models and Bayesian statistics are indicated.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62F15 | Bayesian inference |
Keywords:
Wishart distributions; homogeneous cones; Vinberg algebras; exponential families; posets; matrices with zeroes; normal graphical modelsReferences:
| [1] | Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley, New York. · Zbl 0651.62041 |
| [2] | Anderson, T. W. (1969). Statistical inference for covariance matrices with linear structure. In Proc.Second Internat.Symp.Multivariate Anal. In Krishnaiah, P. R. (ed.), Academic, New York. pp. 55-66. |
| [3] | Anderson, T. W. (1970). Estimation of covariance matrices which are linear combinations or whose inverses are linear combinations of given matrices. In Essays in Probability and Statistics. (Bose, R. C., Chakravati, I. M., Mahalanobis, P. C., Rao, C. R. and Smith, K. J. C. eds.), 1-24. Univ. North Carolina Press, Chapel Hill. · Zbl 0265.62023 |
| [4] | Anderson, T. W. (1973). Asymptotically efficient estimation of covariances with linear structure. Ann.Stat. 1, 135-141. · Zbl 0296.62022 · doi:10.1214/aos/1193342389 |
| [5] | Andersson, S. A. (1975). Forel?sningsnoter i. erdimensional statististisk analyse. Lecture notes. Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish). |
| [6] | Andersson, S. A. (1975). Invariant normal models. Ann.Stat. 3, 132-154. · Zbl 0373.62029 · doi:10.1214/aos/1176343004 |
| [7] | Andersson, S. A. (1976). Forel?sningsnoter i. erdimensional statististisk analyse. Lecture notes, Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish). |
| [8] | Andersson, S. A. (1978). On the mathematical foundation of multivariate statistical analysis. Lecture Notes presented in Lunteren, Holland, Nov. 1978, Inst. Mathematical Statistics. Univ. Copenhagen. |
| [9] | Andersson, S. A. (1992). Normal linear models given by group symmetry. Lecture Notes presented at the DMW-Seminar Guntzburg, Germany. Inst. Mathematical Statistics, Univ. Copenhagen. |
| [10] | Andersson, S. A. (2004). Hypothesis testing for the general Wishart distribution on homogeneous cones. In preparation. |
| [11] | Andersson, S. A., Br?ns, H. K., and Tolver Jensen, S. (1975). En algebraisk teori for normale statistiske modeller. Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish). |
| [12] | Andersson, S. A., Br?ns, H. K., and Tolver Jensen, S. (1983). Distribution of Eigenvalues in multivariate statistical analysis. Ann.Stat. 11, 392-415. · Zbl 0517.62053 · doi:10.1214/aos/1176346149 |
| [13] | Andersson, S. A., Madigan, D., Perlman, M. D., and Triggs, C. M. (1995). On the relation between conditional independence models determined by nite distributive lattice and directed acyclic graphs. J.Stat.Plan.Inference 48, 25-46. · Zbl 0839.62063 · doi:10.1016/0378-3758(94)00150-T |
| [14] | Andersson, S. A., Madigan, D., Perlman, M. D., and Triggs, C. M. (1997). A graphical characterization of lattice conditional independence models. Artif.Intell. 21, 27-50. · Zbl 0888.68090 |
| [15] | Andersson, S. A., and Madsen, J. (1998). Symmetry and lattice conditional independence in a multivariate normal distribution. Ann.Stat. 26, 525-572. · Zbl 0943.62047 · doi:10.1214/aos/1028144848 |
| [16] | Andersson, S. A., and Perlman, M. D. (1993). Lattice models for conditional independence in a multivariate normal distribution. Ann.Stat. 21, 1318-1358. · Zbl 0803.62042 · doi:10.1214/aos/1176349261 |
| [17] | Andersson, S. A., and Perlman, M. D. (1994). Normal linear models with lattice conditional independence restrictions. Multivariate Anal.Appl. 24, 97-110. |
| [18] | Andersson, S. A., Madigan, D., and Perlman, M. D. (2001). Alternative Markov properties for chain graphs. Scand.J.Stat. 28, 33-85. · Zbl 0972.60067 · doi:10.1111/1467-9469.00224 |
| [19] | Andersson, S. A., Letac, G., and Massam, H. (2004). The variance of the Wishart distributions on homogeneous cones. In preparation. |
| [20] | Andersson, S. A., and Wojnar G. G. (2004). The Wishart Distribution on Homogeneous Cones. Acta et Commentstiones Universitatis Tartuensis de Mathematica 8. To appear. · Zbl 1058.62044 |
| [21] | Andersson, S. A., and Sherrer, C. (2004). The multivariate circular symmetry model. In preparation. |
| [22] | Arnold, S. F. (1973). Application of the theory of products of certain patterned covariance matrices. Ann.Stat. 1, 682-699. · Zbl 0289.62038 · doi:10.1214/aos/1176342463 |
| [23] | Baez, J. C. (2002). The octonions. Bull.Amer.Math.Soc. 39 (2)145-205. · Zbl 1026.17001 · doi:10.1090/S0273-0979-01-00934-X |
| [24] | Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory, Wiley, New York. · Zbl 0387.62011 |
| [25] | Bourbaki, N. (1963). ?El ?ements de Math ?ematique. Integration, Ch. 7 ? 8. Hermann, Paris. · Zbl 0156.03204 |
| [26] | Bourbaki, N. (1971). ?El ?ements de Math ?ematique. Topologie G ?en ?erale. Ch. 1 ? 4. Hermann, Paris. · Zbl 0249.54001 |
| [27] | Br?ns, H. K. (1969). Forel?sningsnoter i den normale fordelings teori. Personal communication, Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish). |
| [28] | Br?ns, H. K. (1998). Flerdimensional statistisk analyse. Manuscript. Dept. of Mathematical Statistics, Univ. Copenhagen. (In Danish). |
| [29] | Casalis, M. (1991). Les families exponentielles ?a variance quadratique homog?ne sont de lois de Wishart sur un cone sym ?etrique. C.R.Acad.Sci.Paris S ?er.I Math. 312, 537-540. · Zbl 0745.62051 |
| [30] | Casalis, M., and Letac, G. (1996). The Lukacs-Olkin-Rubin characterization of the Wishart distribution on symmetric cones. Ann.Stat. 24, 763-786. · Zbl 0906.62053 · doi:10.1214/aos/1032894464 |
| [31] | Castelo, R., and Siebes, A. (2003). A characterization of moral transitive acyclic graphs Markov models as labeled trees. J. Stat.Plann.Inference 115, 235-259. · Zbl 1045.62054 · doi:10.1016/S0378-3758(02)00143-X |
| [32] | Faraut, J., and Kor ?anyi, A. (1994). Analysis on Symmetric Cones. Oxford University Press, New York. · Zbl 0841.43002 |
| [33] | Goodman, N. R. (1963). Statistical analysis based on certain multivariate complex distributions. Ann.Math.Stat. 34, 152-176. · Zbl 0122.36903 · doi:10.1214/aoms/1177704250 |
| [34] | Jacobson, N. (1968). Structure and Representations of Jordan Algebras. Amer. Math. Soc., Providence, R. I. · Zbl 0218.17010 |
| [35] | Jensen, J. L. (1991). A large deviation-type approximation for the ?Box class ?of likelihood ratio criteria. J.Amer.Stat.Assoc. 86, 437-440. · doi:10.2307/2290590 |
| [36] | Khatri, C. G. (1965a). Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann.Math.Stat. 36, 98-114. · Zbl 0135.19506 · doi:10.1214/aoms/1177700274 |
| [37] | Khatri, C. G. (1965b). A test of reality of a covariance matrix in certain complex Gaussian distributions. Ann.Math.Stat. 36, 115-119. · Zbl 0135.19505 · doi:10.1214/aoms/1177700275 |
| [38] | Lauritzen, S. L. (1996). Graphical Models. Oxford University Press, Oxford. · Zbl 0907.62001 |
| [39] | Letac, G., and Massam, H. (1998). Quadratic and inverse regression for the Wishart distribution. Ann.Stat. 26, 573-595. · Zbl 1073.62536 · doi:10.1214/aos/1028144849 |
| [40] | Letac, G., and Massam, H. (2000). Representations of the Wishart distributions. Contemp.Math. 261, 121-142. · Zbl 0971.62026 |
| [41] | Massam, H., and Neher, E. (1997). On transformations and determinants of Wishart variables on symmetric cones. J.Theoret.Prob. 10, 867-902. · Zbl 0890.60016 · doi:10.1023/A:1022658415699 |
| [42] | Olkin, I., and Press, S. J. (1969). Testing and estimation for a circular stationary model. Ann.Math.Stat. 40, 1358-1373. · Zbl 0186.51801 · doi:10.1214/aoms/1177697508 |
| [43] | Olkin, I. (1973). Testing and estimation for structures which are circularly symmetric in blocks. Proc.Res.Sem.Multivariate Statistical Inference. Dalhousie.Nova Scotia, North Holland, Amsterdam, pp. 183-195. · Zbl 0261.62055 |
| [44] | Perlman, M. D. (1987). Group symmetry covariance models. Stat.Sci. 2, 421-425. · doi:10.1214/ss/1177013114 |
| [45] | Seely, J. (1971). Quadratic subspaces and completeness. Ann.Math.Stat. 42, 710-721. · Zbl 0249.62067 · doi:10.1214/aoms/1177693420 |
| [46] | Seely, J. (1972). Completeness for a family of multivariate normal distributions. Ann.Math.Stat. 43, 1644-1647. · Zbl 0257.62018 · doi:10.1214/aoms/1177692396 |
| [47] | Sherrer, C. (2003). Multivariate circular symmetry models. Ph.D.Thesis, Indiana University. |
| [48] | Tolver Jensen, S. (1973). Forel?sninger i matematisk statistik. Lecture notes, Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish.) |
| [49] | Tolver Jensen, S. (1974). Forel?sningsnoter. erdimensional statistisk analyse. Lecture notes, Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish.) |
| [50] | Tolver Jensen, S. (1977). Flerdimensional statistisk analyse. Lecture notes, Inst. Mathematical Statistics, Univ. Copenhagen. (In Danish.) |
| [51] | Tolver Jensen, S. (1983). Symmetrimodeller. Lecture notes. Inst. Mathematical Statistics. Univ. Copenhagen. (In Danish.). |
| [52] | Tolver Jensen, S. (1988). Covariance hypotheses which are linear in both the covariance and the inverse covariance. Ann.Stat. 16, 302-322. · Zbl 0653.62042 · doi:10.1214/aos/1176350707 |
| [53] | Vinberg, E. B. (1960). Homogeneous cones. Dokl.Akad.Nauk SSSR. 133, 9-12; Soviet Math.Dokl. 1, 787-790. · Zbl 0143.05203 |
| [54] | Vinberg, E. B. (1962). Automorphisms of homogeneous convex cones. Dokl.Akad.Nauk SSSR. 143, 265-268; Soviet Math.Dokl. 3, 371-374. · Zbl 0196.30501 |
| [55] | Vinberg, E. B. (1963). The theory of convex homogeneous cones. Trudy Moskov.Mat.Obsc. 12, 303-358; Trans.Moscow Math.Soc. 12, 340-403. · Zbl 0138.43301 |
| [56] | Vinberg, E. B. (1965). The structure of the group of automorphisms of a homogeneous convex cone. Trudy Moskov.Mat.Obsc. 13, 56-83; Trans.Moskow Math.Soc. 13, 63-93. · Zbl 0224.17010 |
| [57] | Wishart, J. (1928). The generalized product moment distribution in a sample from a normal multivariate population. Biometrika 20A, 32-52. · JFM 54.0565.02 |
| [58] | Votaw, D. F. (1948). Testing compound symmetry in a normal multivariate distribution. Ann.Math.Stat. 19, 447-473. · Zbl 0033.07903 · doi:10.1214/aoms/1177730145 |
| [59] | Wilks, S. S. (1946). Sample criteria for testing equality of means, equality of variances, and equality of covariances in a normal multivariate distribution. Ann.Math.Stat. 17, 257-281. · Zbl 0063.08259 · doi:10.1214/aoms/1177730940 |
| [60] | Wojnar, G. G. (2000). Generalized Wishart models on convex homogeneous cones, Ph.D.Thesis. Indiana University. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
