Distributions of Ratios: From Random Variables to Random Matrices

ABSTRACT

The ratio*R* of two random quantities is frequently encountered in probability and statistics. But while for unidimensional statistical variables the distribution of *R* can be computed relatively easily, for symmetric positive definite random matrices, this ratio can take various forms and its distribution, and even its definition, can offer many challenges. However, for the distribution of its determinant, Meijer G-function often provides an effective analytic and computational tool, applicable at any division level, because of its reproductive property.

The ratio

KEYWORDS

Matrix Variate, Beta Distribution, Generalized-F Distribution, Ratios, Meijer G-Function, Wishart Distribution, Ratio

Matrix Variate, Beta Distribution, Generalized-F Distribution, Ratios, Meijer G-Function, Wishart Distribution, Ratio

Cite this paper

nullT. Pham-Gia and N. Turkkan, "Distributions of Ratios: From Random Variables to Random Matrices,"*Open Journal of Statistics*, Vol. 1 No. 2, 2011, pp. 93-104. doi: 10.4236/ojs.2011.12011.

nullT. Pham-Gia and N. Turkkan, "Distributions of Ratios: From Random Variables to Random Matrices,"

References

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[10] T. Pham-Gia and N. Turkkan, “The Product and Quotient of General Beta Distribu-tions,” Statistical Papers, Vol. 43, No. 4, 2002, pp. 537-550. doi:10.1007/s00362-002-0122-y

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[14] P. C. B. Phillips, “The Distribution of Matrix Quotients,” Journal of Multivari-ate Analysis, Vol. 16, No. 1, 1985, pp. 157-161. doi:10.1016/0047-259X(85)90056-9

[15] A. M. Mathai, “Jacobians of Matrix Transformations and Functions of Matrix Argument,” World Scientific, Singapore, 1997.

[16] R. J. Muirhead, “Aspects of Multivariate Statistical Theory,” Wiley, New York, 1982. doi:10.1002/9780470316559

[17] A. Kshirsagar, “Multivariate Analysis,” Marcel Dekker, New York, 1972.

[18] I. Olkin and H. Rubin, “Multivariate Beta Distributions and Independence Properties of the Wishart Distribution,” Annals Mathematical Statistics, Vol. 35, No.1, 1964, pp. 261-269. doi:10.1214/aoms/1177703748

[19] M. D. Perlman, “A Note on the Matrix-Variate F Distribution,” Sankhya, Series A, Vol. 39, 1977, pp. 290-298.

[20] T. Pham-Gia, “The Multivaraite Selberg Beta Distribution and Applications,” Statistics, Vol. 43, No. 1, 2009, pp. 65-79.doi:10.1080/02331880802185372

[21] T. Pham-Gia and N. Turkkan, “Distributions of Ratios of Random Variables from the Power-Quadratic Exponential family and Applica-tions,” Statistics, Vol. 39, No. 4, 2005, pp. 355-372.

[22] A. Bekker, J. J. J. Roux and T. Pham-Gia, “Operations on the Matrix Variate Beta Type I Variables and Applications,” Un-published manuscript, University of Pretoria, Pretoria, 2005.

[23] A. M. Mathai, “Extensions of Wilks’ Integral Equations and Distributions of Test Statistics,” Annals of the Institute of Statistical Mathematics, Vol. 36, No. 2, 1984, A, pp. 271-288.

[24] T. Pham-Gia and N. Turkkan, “Exact Expres-sion of the Sample Generalized variance and Applications,” Statistical Papers, Vol. 51, No. 4, 2010, pp. 931-945.

[1] K. C. S. Pillai, “Distributions of Characteristic Roots in Multi-variate Analysis, Part I: Null Distributions,” Communications in Statistics—Theory and Methods, Vol. 4, No. 2, 1976, pp. 157-184.

[2] K. C. S. Pillai, “Distributions of Characteristic Roots in Multivariate Analysis, Part II: Non-Null Distribu-tions,” Communications in Statistics—Theory and Methods, Vol. 5, No. 21, 1977, pp. 1-62.

[3] P. Koev and A. Edelman, “The Effective Evaluation of the Hypergeometric Function of a Matrix Argument,” Mathematics of Computation, Vol. 75, 2006, pp. 833-846. doi:10.1090/S0025-5718-06-01824-2

[4] A. M. Mathai and R. K. Saxena, “Generalized Hypergeometric Functions with Ap-plications in Statistics and Physical Sciences,” Lecture Notes in Mathematics, Vol. 348, Springer-Verlag, New York, 1973.

[5] M. Springer, “The Algebra of Random Variables,” Wiley, New York, 1984.

[6] T. Pham-Gia, “Exact Distribu-tion of the Generalized Wilks’s Distribution and Applications,” Journal of Muo- tivariate Analysis, 2008, 1999, pp. 1698-1716.

[7] V. Adamchik, “The Evaluation of Integrals of Bessel Functions via G-Function Identities,”Journal of Com-putational and Applied Mathematics, Vol. 64, No. 3, 1995, pp. 283-290. doi:10.1016/0377-0427(95)00153-0

[8] T. Pham-Gia, T., N. Turkkan and E. Marchand, “Distribution of the Ratio of Normal Variables,” Communica- tions in Statis-tics—Theory and Methods, Vol. 35, 2006, pp. 1569-1591.

[9] T. Pham-Gia and N. Turkkan, “Distributions of the Ratios of Independent Beta Variables and Applications,” Com- munications in Statistics—Theory and Methods, Vol. 29, No. 12, 2000, pp. 2693-2715. doi:10.1080/03610920008832632

[10] T. Pham-Gia and N. Turkkan, “The Product and Quotient of General Beta Distribu-tions,” Statistical Papers, Vol. 43, No. 4, 2002, pp. 537-550. doi:10.1007/s00362-002-0122-y

[11] T. Pham-Gia and N. Turkkan, “Operations on the Generalized F-Variables, and Applications,” Statistics, Vol. 36, No. 3, 2002, pp. 195-209. doi:10.1080/02331880212855

[12] A. K. Gupta and D. K. Nagar, “Matrix Variate Distributions,” Chapman and Hall/CRC, Boca Raton, 2000.

[13] A. K. Gupta and D. G. Kabe, “The Distribution of Symmetric Matrix Quotients,” Journal of Mul-tivariate Analysis, Vol. 87, No. 2, 2003, pp. 413-417. doi:10.1016/S0047-259X(03)00046-0

[14] P. C. B. Phillips, “The Distribution of Matrix Quotients,” Journal of Multivari-ate Analysis, Vol. 16, No. 1, 1985, pp. 157-161. doi:10.1016/0047-259X(85)90056-9

[15] A. M. Mathai, “Jacobians of Matrix Transformations and Functions of Matrix Argument,” World Scientific, Singapore, 1997.

[16] R. J. Muirhead, “Aspects of Multivariate Statistical Theory,” Wiley, New York, 1982. doi:10.1002/9780470316559

[17] A. Kshirsagar, “Multivariate Analysis,” Marcel Dekker, New York, 1972.

[18] I. Olkin and H. Rubin, “Multivariate Beta Distributions and Independence Properties of the Wishart Distribution,” Annals Mathematical Statistics, Vol. 35, No.1, 1964, pp. 261-269. doi:10.1214/aoms/1177703748

[19] M. D. Perlman, “A Note on the Matrix-Variate F Distribution,” Sankhya, Series A, Vol. 39, 1977, pp. 290-298.

[20] T. Pham-Gia, “The Multivaraite Selberg Beta Distribution and Applications,” Statistics, Vol. 43, No. 1, 2009, pp. 65-79.doi:10.1080/02331880802185372

[21] T. Pham-Gia and N. Turkkan, “Distributions of Ratios of Random Variables from the Power-Quadratic Exponential family and Applica-tions,” Statistics, Vol. 39, No. 4, 2005, pp. 355-372.

[22] A. Bekker, J. J. J. Roux and T. Pham-Gia, “Operations on the Matrix Variate Beta Type I Variables and Applications,” Un-published manuscript, University of Pretoria, Pretoria, 2005.

[23] A. M. Mathai, “Extensions of Wilks’ Integral Equations and Distributions of Test Statistics,” Annals of the Institute of Statistical Mathematics, Vol. 36, No. 2, 1984, A, pp. 271-288.

[24] T. Pham-Gia and N. Turkkan, “Exact Expres-sion of the Sample Generalized variance and Applications,” Statistical Papers, Vol. 51, No. 4, 2010, pp. 931-945.