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Selberg-Type Generalized Quadratic Forms Gamma and Beta Integrals

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2017

Key Words:

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Abstract (2. Language): 
Although Selberg-type single positive de nite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6].
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REFERENCES

References: 

[1] Askey, Richard and Richards, Donald. (1989) Selbergs second beta integral and an
integral of Mehta, Probability, Statistics, and Mathematics. Academic Press, (Karlin
Volume), New York.
[2] Gupta, A.K. and Kabe, D.G. (2005). On Selberg beta integrals Random Obr. And
Stocastic Eqn 13, 11-16.
[3] Gupta, A.K. and Kabe, D.G. (2007). The doubly noncentral multivariate beta distri-
bution. J. Appli. Stat. Sc. 1,21-25.
[4] Gupta, A.K. and Kabe, D.G. (2008). Selberg-type squared matrices gamma and beta
integrals. Eu. J. Pure and Appli. Math. 1,197-201.
[5] Gupta, A.K. and Nagar, D.K. (2007). Matrix Variate Distributions
[6] Kabe, D.G. (1984). Classical statistical analysis based on a certain hypercomplex
multivariate normal distribution. Metrika. 31, 63-76.
[7] Mathai, A.M., Hayakawa, T., Provost, S.B. (1995). Bilinear Forms and Zonal Poly-
nomials. Springer-Verlag, New York.
[8] Mathai, A.M. (1997). Jacobians of Matrix Transformations and Functions of Matrix
Arguments. World Scientic, London, England.

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