平均ベクトルと共分散行列の性質

まず,尤度関数は以下.
$\begin{array}{rl}L(\mathit{\mu },\mathbf{\Sigma })& =\prod _{i=1}^{n}f({\mathit{y}}_{\mathit{i}};\mathit{\mu },\mathbf{\Sigma })\\ & =\prod _{i=1}^n\frac{1}{(\sqrt{2\pi }{)}^p|\mathbf{\Sigma }{|}^{1/2}}exp[-\frac{1}{2}({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}^{-1}({\mathit{y}}_{\mathit{i}}-\mathit{\mu })]\\ & =\frac{1}{(\sqrt{2\pi }{)}^{np}|\mathbf{\Sigma }{|}^{n/2}}exp[-\frac{1}{2}\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}^{-1}({\mathit{y}}_{\mathit{i}}-\mathit{\mu })]\end{array}$

ここで,
$\begin{array}{rl}\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}^{-1}({\mathit{y}}_{\mathit{i}}-\mathit{\mu })& =\sum _{i=1}^{n}tr({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}^{-1}({\mathit{y}}_{\mathit{i}}-\mathit{\mu })\\ & =tr[{\mathbf{\Sigma }}^{-1}\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\mathit{\mu })({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }]\end{array}$
さらに
$\begin{array}{rl}\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\mathit{\mu })({\mathit{y}}_{\mathit{i}}-\mathit{\mu }{)}^{\prime }& =\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\bar{\mathit{y}}+\bar{\mathit{y}}-\mathit{\mu })({\mathit{y}}_{\mathit{i}}-\bar{\mathit{y}}+\bar{\mathit{y}}-\mathit{\mu }{)}^{\prime }\\ & =\sum _{i=1}^n({\mathit{y}}_{\mathit{i}}-\bar{\mathit{y}})({\mathit{y}}_{\mathit{i}}-\bar{\mathit{y}}{)}^{\prime }+n(\bar{\mathit{y}}-\mathit{\mu })(\bar{\mathit{y}}-\mathit{\mu }{)}^{\prime }\\ & =(n-1)\mathit{S}+n(\bar{\mathit{y}}-\mathit{\mu })(\bar{\mathit{y}}-\mathit{\mu }{)}^{\prime }\end{array}$
したがって,

$\begin{array}{rl}L(\mathit{\mu },\mathbf{\Sigma })& =\frac{1}{(\sqrt{2\pi }{)}^{np}|\mathbf{\Sigma }{|}^{n/2}}exp[-\frac{1}{2}\{(n-1)tr{\mathbf{\Sigma }}^{-1}\mathit{S}+n(\bar{\mathit{y}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}^{-1}(\bar{\mathit{y}}-\mathit{\mu })\}]\\ & =g(\bar{\mathit{y}},\mathit{S},\mathit{\mu },\mathbf{\Sigma })\cdot 1\end{array}$

いま, $n\times n$の直交行列で,その第1行が${\mathbf{1}}_{\mathit{n}}^{{}^{\mathbf{\prime }}}/\sqrt{n}$であるような行列$\mathbf{\Gamma }$を考え,
$\mathit{Y}=({\mathit{Y}}_{\mathbf{1}},{\mathit{Y}}_{\mathbf{2}},\dots ,{\mathit{Y}}_{\mathit{n}}{)}^{{}^{\prime }}=\mathbf{\Gamma }\mathit{X}$とする.

このとき, 補題${}^{※}$より${\mathit{Y}}_{\mathbf{1}},{\mathit{Y}}_{\mathbf{2}},\dots ,{\mathit{Y}}_{\mathit{n}}\in {\mathbb{R}}^p$は互いに独立に,共分散行列$\mathbf{\Sigma }$の多変量正規分布に従う.

よって,
${\displaystyle {\mathit{Y}}_{\mathbf{1}}={\mathbf{\Gamma }}_{\mathbf{1}}^{{}^{\prime }}\mathit{X}=\frac{1}{\sqrt{n}}\sum _{i=1}^n{\mathit{X}}_{\mathit{i}}=\sqrt{n}\frac{1}{n}\sum _{i=1}^n{\mathit{X}}_{\mathit{i}}=\sqrt{n}\bar{\mathit{X}}}$
また$i\ge 2$において,$\mathrm{\Gamma }$の直交性より
$E[{\mathit{Y}}_{\mathit{i}}]=E[{\mathbf{\Gamma }}_{\mathit{i}}^{{}^{\prime }}\mathit{X}]={\mathbf{\Gamma }}_{\mathit{i}}^{{}^{\prime }}E[\mathit{X}]={\mathbf{\Gamma }}_{\mathit{i}}^{{}^{\prime }}\mathit{\mu }=0$
が成り立つ.

一方,
$\begin{array}{rl}W& =\sum _{i=1}^n({\mathit{X}}_{\mathit{i}}-\bar{\mathit{X}})({\mathit{X}}_{\mathit{i}}-\bar{\mathit{X}}{)}^{{}^{\prime }}=\sum _{i=1}^n{\mathit{X}}_{\mathit{i}}{\mathit{X}}_{\mathit{i}}^{{}^{\prime }}-n\bar{\mathit{X}}\overline{{\mathit{X}}^{{}^{\prime }}}\\ & ={\mathit{X}}^{{}^{\prime }}\mathit{X}-{\mathit{Y}}_{\mathbf{1}}{\mathit{Y}}_{\mathbf{1}}^{{}^{\prime }}={\mathit{Y}}^{{}^{\prime }}\mathit{Y}-{\mathit{Y}}_{\mathbf{1}}{\mathit{Y}}_{\mathbf{1}}^{{}^{\prime }}\\ & ={\mathit{Y}}_{\mathbf{2}}{\mathit{Y}}_{\mathbf{2}}^{{}^{\prime }}+\cdots +{\mathit{Y}}_{\mathit{n}}{\mathit{Y}}_{\mathit{n}}^{{}^{\prime }}\end{array}$
となり,$W$は${\mathit{Y}}_{\mathbf{1}}$を含まない形で表される.
${\mathit{Y}}_{\mathbf{1}}$と${\mathit{Y}}_{\mathbf{2}},\dots ,{\mathit{Y}}_{\mathit{n}}$の独立性から,${\mathit{Y}}_{\mathbf{1}}$と$\mathit{W}$は独立.

すなわち$\bar{\mathit{X}}$と$\mathit{W}$は独立である.