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

## Main Article Content

## 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.

### Downloads

Download data is not yet available.

## Article Details

How to Cite

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

**Ingeniería y Ciencia | ing.cienc.**, [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: 22 sep. 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.

Issue

Section

Articles

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

## 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