The Wishart distribution plays a fundamental role in multivariate statistical analysis. In this paper, we introduced a generalized Wishart distribution for which the underlying observed vectors follow a Kotz-type distribution. Several properties of the Kotz-Wishart (KW) random matrix and its inverted version are investigated. Unbiased estimator for the parameter matrix is obtained. Explicit forms for the probability density functions (pdf) and the moment generating function (mgf) are derived. Further, inspired by the particular form of the pdf of KW random matrix, we introduced a matrix variate version of a generalized Laplace transform, which is known in the literature as Varma transform [31]. The proposed M-Varma transform is used to extend some results involving special functions of matrix argument. Also, it is shown that an identity established by Herz [15], by means of Laplace transform, remains robust under the M-Varma transform.

Kotz type distribution, Kotz-Wishart distribution, estimation, zonal polynomials, hypergeometric functions, M-Varma transform.

How to cite this article: Amadou Sarr, A generalized Wishart distribution: matrix variate Varma transform, Far East Journal of Theoretical Statistics 63(2) (2021), 51-83. DOI: 10.17654/0972086321001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] S. Adhikari, Generalized Wishart distribution for probabilistic structural dynamics, Comput. Mech. 45 (2010), 495-511.
[2] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed., Wiley, New York, 2003.
[3] J. Bai and S. Shi, Estimating high dimensional covariance matrices and its applications, Ann. Econ. Finance 12(2) (2011), 199-215.
[4] A. Bellahnid and A. Sarr, Skewed Kotz distribution with application to financial stock returns, J. Stat. Theory and Practice 13(4) (2019), 1-33.
doi:10.1007/s42519-019-0054-7.
[5] T. Bodnar, S. Mazur and K. Podgorski, Singular inverse Wishart distribution and its application to portfolio theory, J. Multivariate Anal. 143 (2016), 314-326.
[6] A. G. Constantine, Some non-central distribution problems in multivariate analysis, Ann. Math. Statist. 34 (1963), 1270-1285.
[7] F. J. Caro-Lopera, G. González-Farías and N. Balakrishnan, On generalized Wishart distribution-I: Likelihood ratio test for homogeneity of the covariance matrices, Sankhya A 76 (2014), 179-194.
[8] F. J. Caro-Lopera, J. A. Díaz-García and G. González-Farías, Noncentral elliptical configuration density, J. Multivariate Anal. 101(1) (2010), 32-43.
[9] K. T. Fang and T. W. Anderson, Statistical Inference in Elliptically Contoured and Related Distributions, Allerton Press, New York, 1990.
[10] K. T. Fang, S. Kotz and K. W. Ng, Symmetric Multivariate and Related Distributions, Chapman and Hall, London, New York, 1990.
[11] K. T. Fang and Y. T. Zhang, Generalized Multivariate Analysis, Springer-Verlag, New York, 1990.
[12] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh ed., Academic Press, New York, 2007.
[13] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman and Hall/CRC, Boca Raton, 2000.
[14] A. K. Gupta and T. Varga, Elliptically Contoured Models in Statistics, Kluwer Academic Publishers, Dordrecht, 1993.
[15] C. S. Herz, Bessel functions of matrix argument, Ann. Math. 61 (1955), 474-523.
[16] A. T. James, Zonal polynomials of the real positive definite symmetric matrices, Ann. Math. 74 (1961), 456-469.
[17] H. H. Kelejian and I. R. Prucha, Independent or uncorrelated disturbances in linear regression: an illustration of the difference, Econom. Lett. 19 (1985), 35-38.
[18] P. Koev and A. Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp. 75 (2006), 833-846.
[19] S. Kotz, Multivariate distributions at a cross road, A Modern Course on Statistical Distributions in Scientific Work, Reidel, Dordrecht, 1975, pp. 247-270.
[20] M. O. Kuismin and M. J. Sillanpaa, Estimation of covariance and precision matrix, network structure, and a view toward systems biology, WIRE’s Comput. Stat. 9 (2017), e1415. doi:10.1002/wics.1415.
[21] O. Ledoit and M. Wolf, Improved estimation of the covariance matrix of stock returns with an application to portfolio selection, Journal of Empirical Finance 10 (2004b), 603-621.
[22] J. K. Lindsey, Multivariate elliptically contoured distributions for repeated measurements, Biometrics 55 (1999), 1277-1280.
[23] R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 2005.
[24] S. Nadarajah, The Kotz-type distribution with applications, Statistics 37(4) (2003), 341-358.
[25] F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, Berlin, 1974.
[26] A. Sarr, On a Kotz-Wishart distribution: multivariate Varma transform, 2014. arXiv:1404.4441v1.
[27] A. Sarr and A. K. Gupta, Estimation of the precision matrix of multivariate Kotz type model, J. Multivariate Anal. 100 (2009), 742-752.
[28] A. Sarr and A. K. Gupta, Exponential scale mixture of matrix variate Cauchy distributions, Proc. Amer. Math. Soc. 139(4) (2011), 1483-1494.
[29] H. M. Srivastava, Fractional integration and inversion formulae associated with the generalized Whittaker transform, Pacific J. Math. 26 (1968), 375-377.
[30] B. C. Sutradhar and M. M. Ali, A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model, J. Multivariate Anal. 29 (1989), 155-162.
[31] R. S. Varma, On a generalization of Laplace integral, Proc. Nat. Acad. Sci. India Sect. A 20 (1951), 209-216.
[32] J. Wishart, Generalized product moment distribution in samples from a normal multivariate population, Biometrika 20 (1928), 32-52.