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February, 1968 On the Non-Central Distributions of Two Test Criteria in Multivariate Analysis of Variance
C. G. Khatri, K. C. S. Pillai
Ann. Math. Statist. 39(1): 215-226 (February, 1968). DOI: 10.1214/aoms/1177698521


Let $\mathbf{X}$ be a $p \times f_2$ matrix variate $(p \leqq f_2)$ and $\mathbf{Y}$ a $p \times f_1$ matrix variate $(p \leqq f_1)$ and the columns be all independently normally distributed with covariance matrix $\mathbf{\Sigma}, E(\mathbf({X}) = \mathbf{M}$ and $E(\mathbf{Y}) = \mathbf{0}$. Let $0 < l_1 \leqq \cdots \leqq l_p < 1$ be the ordered characteristic roots of \begin{equation*}\tag{1.1} |\mathbf{XX}' - l(\mathbf{YY}' + \mathbf{XX}')| = \mathbf{0}\end{equation*} and $\omega_1, \cdots, \omega_p$ those of $|\mathbf{MM}' - \omega\mathbf{\Sigma}| = \mathbf{0},$ then the joint density function of $l_1, \cdots, l_p$ is given by Constantine [1], James [2] in the form \begin{equation*}\tag{1.2}\exp (-\frac{1}{2} \operatorname{tr} \mathbf{\Omega}) _1F_1(\frac{1}{2}\nu; \frac{1}{2}f_2; \frac{1}{2}\mathbf{\Omega}, \mathbf{L})f_0(l_1, \cdots, l_p),\end{equation*} where \begin{equation*}\begin{align*}{1.3}f_0(l_1, \cdots, l_p) = C(p, f_1, f_2) \mathbf{\prod}^p_{i=1} \{l_1^{\frac{1}{2}(f_2-p-1)}(1 - l_i)^{\frac{1}{2}(f_1-p-1)}\}\alpha_p (\mathbf{L}), \\ \mathbf{L} = \mathbf{X}'(\mathbf{YY}' + \mathbf{XX}')^{-1}\mathbf{X},\quad \mathbf{\Omega} = \mathbf{M}'\mathbf{\Sigma}^{-1}\mathbf{M},\quad \nu = f_1 + f_2\end{align*},\end{equation*} $_1F_1$ is the hypergeometric function of matrix argument defined in [2] given by $\sum^\infty_{k=0} \sum_\mathbf{kappa}{2}\nu)_\mathbf{kappa}C_\mathbf{kappa}(\frac{1}{2} \mathbf{\Omega})_\mathbf{kappa}(\mathbf{L})/(\frac{1}{2}f_2)\mathbf{kappa} C_\mathbf{kappa}(\mathbf{I}_p)k !$ and where $C(p, f_1, f_2) = \pi^{\frac{1}{2}p^2}\Gamma_p(\frac{1}{2}\nu)/\{\Gamma_p(\frac{1}{2} f_1)\Gamma_p(\frac{1}{2}f_2)\Gamma_p(\frac{1}{2}p)\}$ and $\alpha_p(\mathbf{L} = \mathbf{\prod}_{i>j}(l_i - l_j).$ In this paper the distribution of Pillai's $V^{(p)}$ criterion which is the trace of $\mathbf{L}$, [5], [6] and that of Roy's largest root criterion, $l_p$, [8], [10], have been obtained in series forms and certain constants involved in the series tabulated. In addition, the first four moments of $V^{(p)}$ are also obtained in the linear case, illustrating further use of some of the tabulations.


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C. G. Khatri. K. C. S. Pillai. "On the Non-Central Distributions of Two Test Criteria in Multivariate Analysis of Variance." Ann. Math. Statist. 39 (1) 215 - 226, February, 1968.


Published: February, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0155.27102
MathSciNet: MR219182
Digital Object Identifier: 10.1214/aoms/1177698521

Rights: Copyright © 1968 Institute of Mathematical Statistics


Vol.39 • No. 1 • February, 1968
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