On the Distributions of Direction and Collinearity Factors in Discriminant Analysis

Abstract

Let $A$ and $B$ be two symmetric positive definite matrices of order $p,$ having the independent Wishart densities \begin{equation*}\tag{1.1}g(A) = C_1 \exp \{-\frac{1}{2} \operatorname{tr} A\}|A|^{(n-q-p-1)/2},\end{equation*} where \begin{equation*}\tag{1.2}C_1^{-1} = 2^{(n-q)p/2} \pi^{p(p-1)/4} \mathbf{\prod}^p_{i=1} \Gamma (\frac{1}{2}\lbrack n - q + 1 - i\rbrack),\end{equation*} and \begin{equation*}\tag{1.3}g(B) = C_2 \exp \{-\frac{1}{2} \operatorname{tr} B\}|B|^{(q-p-1)/2} \sum^\infty_{\alpha=0} (\lambda^2b_{11})^\alpha/\lbrack 2^{2\alpha}\alpha!\Gamma(\frac{1}{2}n + \alpha)\rbrack,\end{equation*} where \begin{equation*}\tag{1.4}C^{-1}_2 = \exp \{\frac{1}{2}\lambda^2\}2^{qp/2}\pi^{p(p-1)/4} \mathbf{\prod}^p_{i=2}\Gamma (\frac{1}{2}\lbrack q + 1 - i\rbrack).\end{equation*} Anderson [1] calls the density (1.3) as noncentral linear Wishart density. If we define the matrix $V$ by the relations \begin{equation*}\tag{1.5}A = CVC',\quad A + B = CC',\end{equation*} where $C$ is a lower triangular matrix of order $p.$ Then Kshirsagar [4] finds the noncentral multivariate linear beta density of $V$ to be \begin{equation*}\tag{1.6} g(V) = C_3|V|^{(n-q-p-1)/2}|I - V|^{(q-p-1)/2}\Phi(v_{11}),\end{equation*} where \begin{equation*}\tag{1.7}\Phi(v_{11}) = _1F_1\lbrack\frac{1}{2}n, \frac{1}{2}q; \frac{1}{2}\lambda^2 (1 - v_{11})\rbrack,\end{equation*} and \begin{equation*}\begin{align*}\tag{1.8}C_3 = \exp \{-\frac{1}{2}\lambda^2\} \mathbf{\prod}^p_{i=2} \Gamma(\frac{1}{2}\lbrack n + 1 - i\rbrack)\pi^{-p(p-1)/4} \\ \cdot\lbrack\mathbf{\prod}^p_{i=1} \Gamma(\frac{1}{2}\lbrack n - q + 1 - i\rbrack)\mathbf{\prod}^p_{i=2}\Gamma(\frac{1}{2}\lbrack q + 1 - i\rbrack)\rbrack^{-'}\end{align*}.\end{equation*} Kshirsagar [4] has used this distribution of $V$ to derive the distribution of the test criterion for testing the adequacy of a single hypothetical discriminant function as defined by Williams [6]. Continuing his earlier work, Kshirsagar [5] now uses the distribution of $V$ to obtain the distributions of the direction, and the collinearity factor of this single discriminant function. In case $\xi = \alpha'x$ denotes the discriminant function, then Bartlett [3] gives a factorization of $\Lambda = |V|$ as \begin{equation*}\tag{1.9}\Lambda = \Lambda_1\Lambda_2\Lambda_3,\end{equation*} where \begin{equation*}\begin{align*}\tag{1.10}\Lambda_1 = 1 - (\alpha'B\alpha)/(\alpha'(A + B)\alpha), \\ \Lambda_2 = \lbrack 1 - \alpha'B(A + B)^{-1}B\alpha/\alpha'B\alpha\rbrack/\Lambda_1, \text{direction factor}, \\ \Lambda_3 = \Lambda/\Lambda_1\Lambda_2,\text{the partial collinearity factor}\end{align*}.\end{equation*} Assuming $\alpha = (1, 0, 0, \cdots, 0)$ and factorizing the density of $V$ in terms of rectangular coordinates $T,$ where $V = TT'$ and $T$ lower triangular, Kshirsagar [5] expresses $\Lambda_1, \Lambda_2,$ and $\Lambda_3$ as function of the elements of $T,$ and thus obtains the densities of $\Lambda_1, \Lambda_2,$ and $\Lambda_3.$ Bartlett [3] gives an alternative factorization of $\Lambda$ as $\Lambda = \Lambda_1\Lambda_4\Lambda_5,$ where $\Lambda_4$ is called the collinearity factor and $\Lambda_5$ the partial direction factor, where $\Lambda_4$ and $\Lambda_5$ are certain functions of the elements of the matrices $A$ and $B.$ Kshirsagar expresses $\Lambda_4$ and $\Lambda_5$ as functions of the elements of $T$ and obtains their distributions. Kshirsagar's [5] derivations of these distributions, although elegant, are lengthy and involved, as he uses several lower triangular matrix transformations of the rectangular coordinates in his derivations. It might perhaps be of pedagogical interest to express $\Lambda_1, \Lambda_2, \Lambda_3, \Lambda_4$ and $\Lambda_5$ as functions of the elements of $V$ itself and thus derive their distributions. The present derivations are shorter and neater as they avoid most of the transformations used by Kshirsagar [5]. Since the distributions of $\Lambda$'s are well known beta distributions, they have been derived without the normalizing constants. We assume that all the integrals occurring in this paper are evaluated over appropriate ranges of the variable of integration.