# Partial differential equations for hypergeometric functions of complex argument matrices and their applications

Article

- 94 Downloads
- 5 Citations

## Keywords

Asymptotic Expansion Latent Root Hypergeometric Function Confluent Hypergeometric Function Standard Normal Distribution Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Brillinger, D. R. (1969). The canonical analysis of stationary time series, in
*Multivariate Analysis*-II (P. R. Krishnaiah, Ed.), pp. 331–350. Academic Press, New York.Google Scholar - [2]Chikuse, Y. (1974) Asymptotic expansions for the distributions of the latent roots of two matrices in multivariate analysis, Ph.D. thesis. Yale University.Google Scholar
- [3]Constantine, A. G. and Muirhead, R. J. (1972). Partial differential equations for hypergeometric functions of two argument matrices.
*J. Multivariate Analysis*,**2**, 332–338.CrossRefMathSciNetzbMATHGoogle Scholar - [4]Fujikoshi, Y. (1971) Asymptotic expansions of the non-null distributions of two criteria for the linear hypothesis concerning complex multivariate normal populations,
*Ann. Inst. Statist. Math.*,**23**, 477–490.CrossRefMathSciNetzbMATHGoogle Scholar - [5]Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution,
*Ann. Math. Statist.*,**35**, 152–176.Google Scholar - [6]Goodman, N. R. and Dubman, M. R. (1969). Theory of time-varying spectral analysis and complex Wishart matrix processes, in
*Multivariate Analysis*-II (P. R. Krishnaiah, Ed.), pp. 351–366, Academic Press, New York.Google Scholar - [7]Hayakawa, T. (1972). On the distribution of the latent roots of a complex Wishart matrix (non-central case),
*Ann. Inst. Statist. Math.*,**24**, 1–17.CrossRefMathSciNetzbMATHGoogle Scholar - [8]Hayakawa, T. (1972). The asymptotic distributions of the statistics based on the complex Gaussian distribution,
*Ann. Inst. Statist. Math.*,**24**, 231–244.CrossRefMathSciNetzbMATHGoogle Scholar - [9]Herz, C. S. (1955). Bessel functions of matrix argument,
*Ann. Math.*,**61**, 474–523.CrossRefMathSciNetGoogle Scholar - [10]James, A. T. (1964). Distribution of matrix variates and latent roots derived from normal samples,
*Ann. Math. Statist.*,**35**, 475–501.MathSciNetzbMATHGoogle Scholar - [11]James, A. T. (1968). Calculation of zonal polynomial coefficients by use of the Laplace-Beltrami operator,
*Ann. Math. Statist.*,**39**, 1711–1718.MathSciNetzbMATHGoogle Scholar - [12]Kendall, M. G. and Stuart, A. (1969).
*The Advanced Theory of Statistics*, Vol. 1, Hafner, New York.zbMATHGoogle Scholar - [13]Khatri, C. G. (1964). Distribution of the largest or the smallest characteristics root under null hypothesis concerning complex multivariate normal populations,
*Ann. Math. Statist.*,**35**, 1807–1810.MathSciNetzbMATHGoogle Scholar - [14]Khatri, C. G. (1965), Classical statistical analysis based on a certain multivariate complex Gaussian distribution,
*Ann. Math. Statist.*,**36**, 98–114.MathSciNetzbMATHGoogle Scholar - [15]Khatri, C. G. (1966). On the distribution problems based on positive definite quadratic functions in normal vectors,
*Ann. Math. Statist.*,**37**, 468–479.MathSciNetzbMATHGoogle Scholar - [16]Khatri, C. G. (1969). Non-central distributions of ith largest characteristic roots of three matrices concerning complex multivariate normal populations,
*Ann. Inst. Statist. Math.*,**21**, 23–32.MathSciNetzbMATHGoogle Scholar - [17]Li, H. C., Pillai, K. S. C. and Chang, T. C. (1970). Asymptotic expansions for distributions of the roots of two matrices from classical and complex Gaussian populations.
*Ann. Math. Statist.*,**41**, 1541–1556.MathSciNetzbMATHGoogle Scholar - [18]Muirhead, R. J. (1970). Systems of partial differential equations for hypergeometric functions of matrix argument,
*Ann. Math. Statist.*,**41**, 991–1001.MathSciNetzbMATHGoogle Scholar - [19]Muirhead, R. J. (1970). Asymptotic distributions of some multivariate tests,
*Ann. Math. Statist.*,**41**, 1002–1010.MathSciNetzbMATHGoogle Scholar - [20]Muirhead, R. J. (1972). The asymptotic non-central distribution of Hotelling's generalized
*T*_{0}^{2}*Ann. Math. Statist.*,**43**, 1671–1677.MathSciNetzbMATHGoogle Scholar - [21]Muirhead, R. J. and Chikuse, Y. (1975). Asymptotic expansions for the joint and marginal distributions of the latent roots of the covariance matrix,
*Ann. Statist.*,**3**, 1011–1017.MathSciNetzbMATHGoogle Scholar - [22]Priestley, M. B., Subba Rao, T. and Tong, H. (1973). Identification of the structure of multivariable stochastic systems, in
*Multivariate Analysis*-III (P. R. Krishnaiah, Ed.), pp. 351–368. Academic Press, New York.Google Scholar - [23]Sugiura, N. (1973). Derivatives of the characteristic root of a symmetric or a Hermitian matrix with two applications in multivariate analysis,
*Commun. Statist.*,**1**, 393–417.MathSciNetCrossRefGoogle Scholar - [24]Sugiyama, T. (1972). Distributions of the largest latent root of the multivariate complex Gaussian distribution,
*Ann. Inst. Statist. Math.*,**24**, 87–94.CrossRefMathSciNetzbMATHGoogle Scholar

## Copyright information

© The Institute of Statistical Mathematics 1976