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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

Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
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. 475-501, 1964
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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. Díaz-García, 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), Díaz-García and Gutiérrez-Jáimez (2001) and James (1964)...

### José A. Díaz-García, 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. Diversity-Multiplexing Tradeoff of Double Scattering MIMO Channels

• ...It follows invoking the property of [9] that...

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