# Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications

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## Abstract

In this article, we define and study generalized forms of extended matrix variate gamma and beta functions. By using a number of results from matrix algebra, special functions of matrix arguments and zonal polynomials we derive a number of properties of these newly defined functions. We also give some applications of these functions to statistical distribution theory.

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NAGAR, Daya K.; GÓMEZ-NOGUERA, Sergio Alexander; GUPTA, Arjun K.
Generalized Extended Matrix Variate Beta and Gamma Functions and Their Applications.

**Ingeniería y Ciencia**, [S.l.], v. 12, n. 24, p. 51-82, nov. 2016. ISSN 2256-4314. Available at: <http://publicaciones.eafit.edu.co/index.php/ingciencia/article/view/3404>. Date accessed: 23 nov. 2017. doi: https://doi.org/10.17230/ingciencia.12.24.3.
Keywords

beta function; extended beta function; extended matrix variate beta distri- bution; extended gamma function; gamma function; matrix argument; zonal polynomial.

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## References

[1] A. Bekker, J. J. J. Roux and M. Arashi, “Wishart ratios with dependent structure: new members of the bimatrix beta type IV,” Linear Algebra Appl., vol. 435, no 12, pp. 3243–3260, 2011. 73

[2] A. Bekker, J. J. J. Roux, R. Ehlers, and M. Arashi, “Bimatrix variate beta type IV distribution: relation to Wilks’s statistic and bimatrix variate Kummer-beta type IV distribution,” Comm. Statist. Theory Methods, vol. 40, no. 23, pp. 4165–4178, 2011. 73

[3] A. Bekker, J. J. J. Roux, R. Ehlers, and M. Arashi, “Distribution of the product of determinants of noncentral bimatrix beta variates,” J. Multivariate Anal., vol. 109, pp. 73–87, 2012. 73

[4] Mohamed Ben Farah and Abdelhamid Hassairi, “On the Dirichlet distributions on symmetric matrices,” J. Statist. Plann. Inference, vol. 139, no. 8, pp. 2559–2570, 2009. 54

[5] Taras Bodnar, Stepan Mazur, Yarema Okhrin, “On the exact and approximate distributions of the product of a Wishart matrix with a normal vector,” J. Multivariate Anal., vol. 122, pp. 70–81, 2013. 73

[6] M. Aslam Chaudhry, Asghar Qadir, M. Rafique and S. M. Zubair, “Extension of Euler’s beta function,” J. Comput. Appl. Math., vol. 78, no. 1, pp. 19–32,1997. 53

[7] M. A. Chaudhry and S. M. Zubair, “Generalized incomplete gamma functions with applications,” J. Comput. Appl. Math., vol. 55, pp. 303–324, 1994. 53

[8] M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions with applications. Boca Raton: Chapman & Hall/CRC, 2002. 53

[9] A. G. Constantine, “Some noncentral distribution problems in multivariate analysis,” Ann. Math. Statist., vol. 34, pp. 1270–1285, 1963. 55, 56, 58

[10] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions. Boca Raton: Chapman & Hall/CRC, 2000. 54, 56

[11] A. K. Gupta and D. K. Nagar, “Matrix-variate Gauss hypergeometric distribution,” J. Aust. Math. Soc., vol. 92, no. 3, pp. 335–355, 2012. 54

[12] A. Hassairi and O. Regaig, “Characterizations of the beta distribution on symmetric matrices,” J. Multivariate Anal., vol. 100, no. 8, pp. 1682–1690, 2009. 54

[13] Carl S. Herz, “Bessel functions of matrix argument,” Ann. of Math. (2), vol. 61, no 2, 474–523, 1955. 56

[14] Anis Iranmanesh, M. Arashi, D. K. Nagar and S. M. M. Tabatabaey, “On inverted matrix variate gamma distribution,” Comm. Statist. Theory Methods, vol. 42, no. 1, pp. 28–41, 2013. 73

[15] A. T. James, “Distributions of matrix variate and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964. 55, 56

[16] E. Krätzel, Integral transformations of Bessel-type, Generalized functions and operational calculus (proceedings of the Conference on Generalized Functions and Operational Calculus, Varna, September 29-October 6, 1975, pp. 148–155, Bulgarian Academy of Sciences, Sofia, 1979. 53

[17] C. G. Khatri, “On certain distribution problems based on positive definite quadratic functions in normal vectors,” Ann. Math. Statist., vol. 37, no. 2, pp. 468–479, 1966. 58

[18] Robb J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1982. 54

[19] Daya K. Nagar, Arjun K. Gupta and Luz Estela Sánchez, “A class of integral identities with Hermitian matrix argument,” Proc. Amer. Math. Soc., vol. 134, no. 11, pp. 3329–3341, 2006. 54

[20] Daya K. Nagar, Alejandro Roldán-Correa and Arjun K. Gupta, “Extended matrix variate gamma and beta functions,” J. Multivariate Anal., vol. 122, pp. 53–69, 2013. 54, 67, 69, 70, 79

[21] Daya K. Nagar and Alejandro Roldán-Correa, “Extended matrix variate beta distributions,” Progress in Applied Mathematics, vol. 6, no. 1, pp. 40–53, 2013. 54, 79

[22] Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta, “Extended matrix variate hypergeometric functions and matrix variate distributions,” Int. J. Math. Math. Sci., vol. 2015, Article ID 190723, 15 pages, 2015. 54

[23] Ingram Olkin, “A class of integral identities with matrix argument,” Duke Math. J., vol. 26, pp. 207–213, 1959. 54

[24] Emine Özergin, Mehmet Ali Özarslan and Abdullah Altın, “Extension of gamma, beta and hypergeometric functions,” J. Comput. Appl. Math., vol. 235, pp. 4601–4610, 2011. 53

[25] Raoudha Zine, “On the matrix-variate beta distribution,” Comm. Statist. Theory Methods, vol. 41, no. 9, pp. 1569–1582, 2012. 54

[2] A. Bekker, J. J. J. Roux, R. Ehlers, and M. Arashi, “Bimatrix variate beta type IV distribution: relation to Wilks’s statistic and bimatrix variate Kummer-beta type IV distribution,” Comm. Statist. Theory Methods, vol. 40, no. 23, pp. 4165–4178, 2011. 73

[3] A. Bekker, J. J. J. Roux, R. Ehlers, and M. Arashi, “Distribution of the product of determinants of noncentral bimatrix beta variates,” J. Multivariate Anal., vol. 109, pp. 73–87, 2012. 73

[4] Mohamed Ben Farah and Abdelhamid Hassairi, “On the Dirichlet distributions on symmetric matrices,” J. Statist. Plann. Inference, vol. 139, no. 8, pp. 2559–2570, 2009. 54

[5] Taras Bodnar, Stepan Mazur, Yarema Okhrin, “On the exact and approximate distributions of the product of a Wishart matrix with a normal vector,” J. Multivariate Anal., vol. 122, pp. 70–81, 2013. 73

[6] M. Aslam Chaudhry, Asghar Qadir, M. Rafique and S. M. Zubair, “Extension of Euler’s beta function,” J. Comput. Appl. Math., vol. 78, no. 1, pp. 19–32,1997. 53

[7] M. A. Chaudhry and S. M. Zubair, “Generalized incomplete gamma functions with applications,” J. Comput. Appl. Math., vol. 55, pp. 303–324, 1994. 53

[8] M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions with applications. Boca Raton: Chapman & Hall/CRC, 2002. 53

[9] A. G. Constantine, “Some noncentral distribution problems in multivariate analysis,” Ann. Math. Statist., vol. 34, pp. 1270–1285, 1963. 55, 56, 58

[10] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions. Boca Raton: Chapman & Hall/CRC, 2000. 54, 56

[11] A. K. Gupta and D. K. Nagar, “Matrix-variate Gauss hypergeometric distribution,” J. Aust. Math. Soc., vol. 92, no. 3, pp. 335–355, 2012. 54

[12] A. Hassairi and O. Regaig, “Characterizations of the beta distribution on symmetric matrices,” J. Multivariate Anal., vol. 100, no. 8, pp. 1682–1690, 2009. 54

[13] Carl S. Herz, “Bessel functions of matrix argument,” Ann. of Math. (2), vol. 61, no 2, 474–523, 1955. 56

[14] Anis Iranmanesh, M. Arashi, D. K. Nagar and S. M. M. Tabatabaey, “On inverted matrix variate gamma distribution,” Comm. Statist. Theory Methods, vol. 42, no. 1, pp. 28–41, 2013. 73

[15] A. T. James, “Distributions of matrix variate and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964. 55, 56

[16] E. Krätzel, Integral transformations of Bessel-type, Generalized functions and operational calculus (proceedings of the Conference on Generalized Functions and Operational Calculus, Varna, September 29-October 6, 1975, pp. 148–155, Bulgarian Academy of Sciences, Sofia, 1979. 53

[17] C. G. Khatri, “On certain distribution problems based on positive definite quadratic functions in normal vectors,” Ann. Math. Statist., vol. 37, no. 2, pp. 468–479, 1966. 58

[18] Robb J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1982. 54

[19] Daya K. Nagar, Arjun K. Gupta and Luz Estela Sánchez, “A class of integral identities with Hermitian matrix argument,” Proc. Amer. Math. Soc., vol. 134, no. 11, pp. 3329–3341, 2006. 54

[20] Daya K. Nagar, Alejandro Roldán-Correa and Arjun K. Gupta, “Extended matrix variate gamma and beta functions,” J. Multivariate Anal., vol. 122, pp. 53–69, 2013. 54, 67, 69, 70, 79

[21] Daya K. Nagar and Alejandro Roldán-Correa, “Extended matrix variate beta distributions,” Progress in Applied Mathematics, vol. 6, no. 1, pp. 40–53, 2013. 54, 79

[22] Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta, “Extended matrix variate hypergeometric functions and matrix variate distributions,” Int. J. Math. Math. Sci., vol. 2015, Article ID 190723, 15 pages, 2015. 54

[23] Ingram Olkin, “A class of integral identities with matrix argument,” Duke Math. J., vol. 26, pp. 207–213, 1959. 54

[24] Emine Özergin, Mehmet Ali Özarslan and Abdullah Altın, “Extension of gamma, beta and hypergeometric functions,” J. Comput. Appl. Math., vol. 235, pp. 4601–4610, 2011. 53

[25] Raoudha Zine, “On the matrix-variate beta distribution,” Comm. Statist. Theory Methods, vol. 41, no. 9, pp. 1569–1582, 2012. 54