7.5 Sample Values

In sample matrices for the discriminate in (9) are \begin {equation} \tag {10} \widehat {B} = \sum ^g_{i=1}\big (\underline {X}_i-\overline {\underline {X}}\big )\big (\underline {\overline {X}_i}-\overline {\underline {X}}\big )' \end {equation} and the sample estimate for \(W\) is \begin {equation} \tag {11} \widehat {W} = \sum ^g_{i=1}(n_i-1)S_i = \sum ^g_{i=1}\sum ^{n_i}_{j=1}\big (\underline {X}_{ij}-\overline {\underline {X}_i}\big )\big (\underline {X}_{ij}-\underline {X}_i\big )' \end {equation}

So that \(\displaystyle {S_{pooled}=\frac {\widehat {W}}{n_1+n_2+\cdots +n_g-g}}\qquad (12)\)

\(\widehat {W}\) is an estimate of \(\displaystyle {\Sigma }\). It can be shown that the vector \(\underline {a}\) is the eigenvector of \(\widehat {W}^{-1}\widehat {B}\).

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