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Asymptotic Expansion
Canonical Correlation
Correlation Coefficient
Covariance Matrices
Covariance Matrix
hypergeometric function
laguerre polynomial
Multivariate Distribution
Normal Distribution
Quadratic Form
Zonal Polynomial
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Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
Distributions of Matrix Variates and Latent Roots Derived from Normal Samples,10.1214/aoms/1177703550,The Annals of Mathematical Statistics,Alan T. Ja
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Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
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Citations: 548
)
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Alan T. James
The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii)
canonical correlation
coefficients. The modifications required for the corresponding distributions derived from the complex
normal distribution
are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the
asymptotic expansion
of $_0F_0$ in Section 12. Section 13 has some formulae for moments.
Journal:
The Annals of Mathematical Statistics
, vol. 35, no. 1964, pp. 475501, 1964
DOI:
10.1214/aoms/1177703550
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Citation Context
(246)
...polynomials, extending (
James 1964,
Eq. (22))...
...2. This follows by expanding the exponentials in series of powers and by applying (22) and (27) from
James (1964)
...
José A. DíazGarcía
,
et al.
Generalised shape theory via SV decomposition I
...For properties and applications of zonal and invariant polynomials we refer to Constantine (
1963
), James (
1964
), Davis (
1979
)Davis (
1980
, Chikuse (
1981
), Gupta and Nagar (
2000
), and Nagar and Gupta (
2002
)...
Arjun K. Gupta
,
et al.
Some Bimatrix Beta Distributions
...Gupta and Nagar 2000; Srivastava and Khatri 1979), two definitions have been proposed for each of these, see Olkin and Rubin (1964), Srivastava (1968), DíazGarcía and GutiérrezJáimez (2001) and
James (1964)
...
José A. DíazGarcía
,
et al.
Noncentral bimatrix variate generalised beta distributions
...Theorem 1 ([
5
]–[8]): Let be a central complex Wishart matrix , where the eigenvalues of are distinct and their ordered values are . Let be the ordered positive eigenvalues of with . The joint p.d.f...
Sheng Yang
,
et al.
DiversityMultiplexing Tradeoff of Double Scattering MIMO Channels
...It follows invoking the property of [
9
] that...
Esra Erten
,
et al.
A polarimetric temporal scene parameter and its application to change ...
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Citations
(548)
Generalised shape theory via SV decomposition I
José A. DíazGarcía
,
Francisco J. CaroLopera
Journal:
Metrika
, pp. 125, 2012
Some Bimatrix Beta Distributions
Arjun K. Gupta
,
Daya K. Nagar
Journal:
Communications in Statisticstheory and Methods  COMMUN STATISTTHEOR METHOD
, vol. 41, no. 5, pp. 869879, 2012
Noncentral bimatrix variate generalised beta distributions
(
Citations: 1
)
José A. DíazGarcía
,
Ramón GutiérrezJáimez
Journal:
Metrika
, vol. 73, no. 3, pp. 317333, 2011
DiversityMultiplexing Tradeoff of Double Scattering MIMO Channels
(
Citations: 1
)
Sheng Yang
,
JeanClaude Belfiore
Journal:
IEEE Transactions on Information Theory  TIT
, vol. 57, no. 4, pp. 20272034, 2011
A polarimetric temporal scene parameter and its application to change detection
Esra Erten
,
Olga Chesnokova
,
Cristian Rossi
,
Irena Hajnsek
Conference:
Geoscience and Remote Sensing IEEE International Symposium  IGARSS
, pp. 10911094, 2011