Scientific Research

An Academic Publisher

Trace of the Wishart Matrix and Applications

**Author(s)**Leave a comment

The trace of a Wishart matrix, either central or non-central, has important roles in various multi-variate statistical questions. We review several expressions of its distribution given in the literature, establish some new results and provide a discussion on computing methods on the distribution of the ratio: the largest eigenvalue to trace.

KEYWORDS

Cite this paper

Pham-Gia, T. , Thanh, D. and Phong, D. (2015) Trace of the Wishart Matrix and Applications.

*Open Journal of Statistics*,**5**, 173-190. doi: 10.4236/ojs.2015.53021.

[1] | Tourneret, J.Y., Ferrari, A. and Letac, G. (2005) The Non-Central Wishart Distribution: Properties and Application to Speckle Imaging. 2005 IEEE/SP 13th Workshop on Statistical Signal Processing, Novosibirsk, 17-20 July 2005, 924-929. |

[2] | Díaz-García, J.A. and Gutiérrez-Jáimez, R. (2011) On Wishart Distribution: Some Extensions. Linear Algebra and its Applications, 435, 1296-1310. |

[3] | Pham-Gia, T. and Turkkan, N.T. (2011) Distributions of Ratios: from Random Variables to Random Matrices. Open Journal of Statistics, 1, 93-104. |

[4] |
Chamayou, J.F. and Wesolowski, J. (2009) Lauricella and Humbert Functions through Probabilistic Tools. Integral transforms and Special Functions, 20, 529-538.
http://dx.doi.org/10.1080/10652460802645750 |

[5] | Mathai, A.M. and Pederzoli, G. (1995) Hypergeometric Functions of Many Matrix Variables and Distributions of Generalized Quadratic Forms. American Journal of Mathematical and Management Sciences, 15, 343-354. |

[6] |
Pham-Gia, T. and Turkkan, T.N. (1999) System Availability in a Gamma Alternating Renewal Process. Naval Research Logistics, 46, 822-844. http://dx.doi.org/10.1002/(SICI)1520-6750(199910)46:7<822::AID-NAV5>3.0.CO;2-D |

[7] | Gelfand, I.M., Kapronov, W.M. and Zelevinsky, A.V. (1991) Hypergeometric Functions, Toric Varieties and Newton Polyhedra. In: Kashiwara, I.M. and Miwa, T., Eds., ICM 90 Satellite Conference Proceedings, Springer-Verlag, Tokyo, 104-121. |

[8] |
Aomoto, K. and Kita, M. (2011) Theory of Hypergeometric Functions. Springer, New York.
http://dx.doi.org/10.1007/978-4-431-53938-4 |

[9] |
Saito, M., Sturmfels, B. and Takayama, N. (2000) Grobner Deformations of Hypergeometric Differential Equations. Springer, New York.
http://dx.doi.org/10.1007/978-3-662-04112-3 |

[10] |
Pham-Gia, T. (2008) Exact Distribution of the Generalized Wilks’s statistic and Applications. Journal of Multivariate Analysis, 99, 1698-1716.
http://dx.doi.org/10.1016/j.jmva.2008.01.021 |

[11] | Muirhead, R. (1982) Aspects of Multivariate Statistical Analysis. Wiley, New York. |

[12] |
Mathai, A.M. and Pillai, K.C.S. (1982) Further Results on the Trace of a Non-Central Wishart Matrix. Communications in Statistics—Theory and Methods, 11, 1077-1086.
http://dx.doi.org/10.1080/03610928208828294 |

[13] |
Kotz, S. and Srinivasan, R. (1969) Distribution of Product and Quotient of Bessel Function Variates. Annals of the Institute of Statistical Mathematics, 21, 201-210.
http://dx.doi.org/10.1007/BF02532244 |

[14] | Laha, R.G. (1954) On Some Properties of the Bessel Function Distributions. Bulletin of the Calcutta Mathematical Society, 46, 59-71. |

[15] | Gupta, A.K. and Nagar, D.K. (2000) Matrix Variate Distributions. Chapman and Hall/CRC, Boca Raton. |

[16] | Anderson, T.W. (1982) An Introduction to Multivariate Statistical Analysis. John Wiley and Sons, New York. |

[17] | Mathai, A.M. (1980) Moments of the Trace of a Non-Central Wishart Matrix. Communications in Statistics—Theory and Methods, 9, 795-801. |

[18] |
Ruben, J. (1962) Probability Content of Regions under Spherical Normal Distribution, IV: The Distribution of Homogeneous and Non-Homogeneous Quadratic Functions in Normal Variables. The Annals of Mathematical Statistics, 33, 542-570.
http://dx.doi.org/10.1214/aoms/1177704580 |

[19] | Press, S.J. (1966) Linear Combinations of Non-Central Chi-Square Variates. The Annals of Mathematical Statistics, 37, 480-487. |

[20] | Hartville, D.A. (1971) On the Distribution of Linear Combinations of Non Central Chi Squares. The Annals of Mathematical Statistics, 42, 809-811. |

[21] |
Provost, S. and Ruduik, E. (1996) The Exact Distribution of Indefinite Quadratic Forms in Noncentral Normal Vectors. Annals of the Institute of Statistical Mathematics, 48, 381-394.
http://dx.doi.org/10.1007/BF00054797 |

[22] | Castano-Martinez, A. and Lopez-Blaquez, F. (2006) Distribution of a Sum of Weighted Non Central Chi-Square Variables. TEST, 14, 397-115. |

[23] | Kourouklis, S. and Moschopoulos, P.G. (1985) On the Distribution of the Trace of a Non-Central Wishart Matrix. Metron, 43, 85-92. |

[24] |
Glueck, D.H. and Muller, K.E. (1998) On the Trace of a Wishart. Communications in Statistics—Theory and Methods, 27, 2137-2141.
http://dx.doi.org/10.1080/03610929808832218 |

[25] | De Waal, D.J. (1972) On the Expected Values of the Elementary Symmetric Functions of a Non-Central Wishart Matrix. The Annals of Mathematical Statistics, 43, 344-347. |

[26] | Saw, J.G. (1973) Expectation of Elementary Symmetric Functions of a Wishart Matrix. Annals of Statistics, 1, 580-582. |

[27] | Shah, B.K. and Khatri, C.G. (1974) Proofs of Conjectures about the Expected Values of the Elementary Symmetric Functions of a Non-Central Wishart Matrix. Annals of Statistics, 2, 833-636. |

[28] | Letac, G. and Massam, H. (2004) All Invariant Moments of the Wishart Distribution. Scandinavian Journal of Statistics, 31, 295-318. |

[29] |
Glaser, R.E. (1976) The Ratio of the Geometric Mean to the Arithmetic Mean from a Random Sample from a Gamma Distribution. Journal of the American Statistical Association, 71, 480-487.
http://dx.doi.org/10.1080/01621459.1976.10480373 |

[30] |
Cheng, J., Wag, N. and Tellambura, C. (2010) Probability Density Function of Logarithmic Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power. Proceedings of the 25th Biennial Symposium on Communications, Kingston, 12-14 May 2010, 348-351.
http://dx.doi.org/10.1109/BSC.2010.5472954 |

[31] |
Mauchly, J.W. (1940) Significance Test for Sphericity of a Normal n-Variate Distribution. The Annals of Mathematical Statistics, 11, 204-209.
http://dx.doi.org/10.1214/aoms/1177731915 |

[32] | Pham-Gia, T. and Turkkan, T.N. (2010) Testing Sphericity Using Small Samples. Statistics, 44, 601-616. |

[33] | Glaser, R.E. (1980) A Characterization of Bartlett’s Statistic Involving Incomplete Beta Functions. Biometrika, 67, 53-58. |

[34] |
Gleser, L. (1966) A Note on Sphericity Test. The Annals of Mathematical Statistics, 37, 464-467.
http://dx.doi.org/10.1214/aoms/1177699529 |

[35] | Khatri, C.G. and Srivastava, M.S. (1971) On Exact Non-Null Distributions of Likelihood Ratio Criteria for Sphericity Test and Equality of Two Covariance Matrices. Sankhyā, 33, 201-206. |

[36] | Muller, K.E. and Barton, C.N. (1989) Approximate Power for Repeated-Measures ANOVA Lacking Sphericity. Journal of the American Statistical Association, 84, 549-555. |

[37] | Mathai, A.M. and Tan, W.Y. (1977) The Non-Null Distribution of the Likelihood Ratio Criterion for Testing the Hypothesis That the Covariance Matrix Is Diagonal. Canadian Journal of Statistics, 5, 63-74. |

[38] |
Krishnaiah, P.R. and Shurmann, F.J. (1974) On the Evaluation of Some Distributions That Arise in Simultaneous Tests for the Equality of the Latent Roots of the Covariance Matrix. Journal of Multivariate Analysis, 4, 265-282.
http://dx.doi.org/10.1016/0047-259X(74)90033-5 |

[39] | Nadler, B. (2011) On the Distribution of the Ratio of the Largest Eigenvalue to the Trace of a Wishart Matrix. Unpublished Manuscript on the Internet. |

[40] | Johnstone, I. (2001) On the Distribution of the Largest Eigenvalue in Principal Components Analysis. Annals of Statistics, 29, 295-327. |

[41] |
Troskie, C.G. and Conradie, W.J. (1986) The Distribution of the Ratios of the Characteristic Roots (Condition Numbers) and Their Applications in Principal Component or Ridge Regression. Linear Algebra and Its Applications, 82, 255-279.
http://dx.doi.org/10.1016/0024-3795(86)90156-4 |

[42] |
Davis, A.W. (1972) On the Ratios of the Individual Latent Roots to the Trace of a Wishart Matrix. Journal of Multivariate Analysis, 2, 440-443.
http://dx.doi.org/10.1016/0047-259X(72)90037-1 |

[43] | Pham-Gia, T. and Turkkan, T.N. (2009) Testing a Covariance Matrix: Exact Null Distribution of Its Likelihood Criterion. Journal of Statistical Computation and Simulation, 79, 1331-1340. |

Copyright © 2018 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.