Matrix-variate distribution theory under elliptical models. IV: Joint distribution of latent roots of covariance matrix and the largest and smallest latent roots. (English) Zbl 1333.62143
The object of this paper is a generalization of the classical work on distributions of the latent roots of a sample covariance matrix under the case of elliptical models. The author derives the distributions of the largest and smallest latent roots from a sample covariance matrix under an elliptical model.
In the same time he also “corrects some erroneous results concerning zonal polynomials present in the literature”.
In the same time he also “corrects some erroneous results concerning zonal polynomials present in the literature”.
Reviewer: N. G. Gamkrelidze (Moskva)
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62E15 | Exact distribution theory in statistics |
| 60E05 | Probability distributions: general theory |
References:
| [1] | Caro-Lopera, F. J.; Díaz-García, J. A.; González-Farías, G., Noncentral elliptical configuration density, J. Multivariate Anal., 101, 32-43 (2009) · Zbl 1177.62069 |
| [2] | Caro-Lopera, F. J.; González-Farías, G.; Balakrishnan, N., On generalized Wishart distributions—I: Likelihood ratio test for homogeneity of covariance matrices, Sankhyā Ser. A, 76, 179-194 (2014) · Zbl 1314.62048 |
| [3] | Caro-Lopera, F. J.; González-Farías, G.; Balakrishnan, N., On generalized Wishart distributions—II: Sphericity test, Sankhyā Ser. A, 76, 195-218 (2014) · Zbl 1314.62049 |
| [5] | Constantine, A. G., Noncentral distribution problems in multivariate analysis, Ann. Math. Statist., 34, 1270-1285 (1963) · Zbl 0123.36801 |
| [6] | Díaz-García, J. A.; Gutiérrez-Jáimez, R., Zonal polynomials: A note, Chil. J. Statist., 2, 21-30 (2010) · Zbl 1213.62092 |
| [7] | Díaz-García, J. A.; Gutiérrez-Jáimez, R., Distributions of the compound and scale mixture of vector and spherical matrix variate elliptical distributions, J. Multivariate Anal., 102, 143-152 (2011) · Zbl 1206.62103 |
| [8] | Díaz-García, J. A.; Gutiérrez-Jáimez, R., Some comments on zonal polynomials and their expected values with respect to elliptical distributions, Acta Math. Appl. Sin. Engl. Ser. (2011) · Zbl 1430.62148 |
| [9] | Fang, K. T.; Zhang, Y. T., Generalized Multivariate Analysis (1990), Science Press, Springer-Verlag: Science Press, Springer-Verlag Beijing · Zbl 0724.62054 |
| [10] | Hashiguchi, H.; Niki, N., Numerical computation on distributions of the largest and the smallest latent roots of the Wishart matrix, J. Japanese Soc. Comput. Statist., 19, 1, 45-56 (2006) · Zbl 1327.65091 |
| [11] | James, A. T., The distribution of the latent roots of the covariance matrix, Ann. Math. Statist., 31, 151-158 (1960) · Zbl 0201.52401 |
| [12] | James, A. T., Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist., 39, 1711-1718 (1964) |
| [13] | Khatri, C. G., On certain distribution problems based on positive definite quadratic functions in normal vectors, Ann. Math. Statist., 37, 468-479 (1966) · Zbl 0138.14502 |
| [14] | Khatri, C. G., On the exact finite series distribution of the smallest or the largest root of matrices in three situations, J. Multivariate Anal., 2, 201-207 (1972) · Zbl 0271.62060 |
| [15] | Koev, P.; Edelman, A., The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp., 75, 833-846 (2006) · Zbl 1117.33007 |
| [16] | Li, R., The expected values of invariant polynomials with matrix argument of elliptical distributions, Acta Math. Appl. Sin., 13, 64-70 (1997) · Zbl 0877.62048 |
| [17] | Li, F.; Xue, Y., Zonal polynomials and hypergeometric functions of quaternion matrix argument, Comm. Statist. Theory Methods, 38, 1184-1206 (2009) · Zbl 1167.62415 |
| [18] | Muirhead, R. J., Aspects of Multivariate Statistical Theory (1982), John Wiley & Sons: John Wiley & Sons New York · Zbl 0556.62028 |
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.
