7.3 Fishers Rule Through Mahalanobie’s Distance
Let \((\pi _1)\) and \((\pi _2)\) be two populations and \(\overline {\underline {X}}_1\) and \(\overline {\underline {X}}_2\) be sample mean vectors respectively. Thus, let \(S\) be the common sample covariance. Let \(X\in \mathbb {R}^p\) assign
- 1.
- \(\underline {X}\) to \(\pi _1\) if \(\displaystyle {\big (\overline {X}_1 -\underline {X}\big )'S^{-1}\big (\overline {X}_1-\underline {X}\big ) < \big (\overline {X}_2-\underline {X}\big )'S^{-1}\big (\overline {X}_2-\underline {X}\big )}\) \(\qquad \text {or}\quad d\big (\underline {X},\overline {X}_1\big ) < d\big (\underline {X},\overline {X}_2\big )\)
- 2.
- \(\underline {X}\) to \(\pi _2\) otherwise.
Now, \[\big (\overline {X}_1 - X\big )'S^{-1}\big (\overline {X}_1- X\big ) < \big (\overline {X}_2-X\big )'S^{-1}\big (\overline {X}_2 - X\big )\]
\[\implies \quad \big (\overline {X}_1 -\overline {X}_2 + \overline {X}_2 - X\big )'S^{-1}\big (\overline {X}_1 - X\big ) < \big (\overline {X}_2 - \overline {X}_1 + \overline {X}_1 - X\big ) S^{-1}\big (\overline {X}_2-X\big )\]
\[\big (\overline {X}_1-\overline {X}_2\big )'S^{-1}\big (\overline {X}_1-X\big ) + \big (\overline {X}_2-X\big )'S^{-1}\big (\overline {X}_1-X\big ) < \big (\overline {X}_2-\overline {X}_1\big )'S^{-1}\big (\overline {X}_2-X\big ) + \big (\overline {X}_1-X\big )'S^{-1}\big (\overline {X}_2-X\big )\]
\[\implies \quad \big (\overline {X}_1-\overline {X}_2\big )'S^{-1}\big (\overline {X}_1-X\big ) + \big (\overline {X}_1-\overline {X}_2\big ) S^{-1}\big (\overline {X}-X\big ) <0\]
\[\implies \quad \big (\overline {X}_1-\overline {X}_2\big )'S^{-1}\big (\overline {X}_1+\overline {X}_2-2X\big ) < 0\]
\[\implies \quad \big (\overline {X}_1-\overline {X}_2\big )'S^{-1}\Bigg (\frac {\overline {X}_1+\overline {X}_2}{2}-X\Bigg ) < 0\]
\[\implies \quad \big (\overline {X}_1-\overline {X}_2\big )'S^{-1}\Bigg (\frac {\overline {X}_1+\overline {X}_2}{2}\Bigg ) - \big (\overline {X}_1-\overline {X}_2\big )'S^{-1}X <0\]
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