標本母集団平均対数リスク比の漸近分布

多項分布の周辺分布であることから、 \begin{gather} e \sim \mathrm{B} \left(N,\pi_{11}\right)\\ f \sim \mathrm{B} \left(N,\pi_{12}\right)\\ g \sim \mathrm{B} \left(N,\pi_{21}\right)\\ e+f=n_{1\bullet } \sim \mathrm{B} \left(N,\pi_{1\bullet }\right)\\ e+g=n_{\bullet 1} \sim \mathrm{B} \left(N,\pi_{\bullet 1}\right) \end{gather} 二項分布の正規近似により、標本比率は漸近的に \begin{gather} p_{11} \sim \mathrm{N} \left[\pi_{11},\frac{\pi_{11} \left(1-\pi_{11}\right)}{N}\right]\\ p_{12} \sim \mathrm{N} \left[\pi_{12},\frac{\pi_{12} \left(1-\pi_{12}\right)}{N}\right]\\ p_{21} \sim \mathrm{N} \left[\pi_{21},\frac{\pi_{21} \left(1-\pi_{21}\right)}{N}\right]\\ p_{1\bullet } \sim \mathrm{N} \left[\pi_{1\bullet },\frac{\pi_{1\bullet } \left(1-\pi_{1\bullet }\right)}{N}\right]\\ p_{\bullet 1} \sim \mathrm{N} \left[\pi_{\bullet 1},\frac{\pi_{\bullet 1} \left(1-\pi_{\bullet 1}\right)}{N}\right]\\ \end{gather} ここで、和の分散の公式より、 \begin{align} V \left(p_{12}-p_{21}\right)&=V \left(p_{12}\right)+V \left(p_{21}\right)-2\mathrm{Cov} \left(p_{12},p_{21}\right)\\ &=\frac{\pi_{12} \left(1-\pi_{12}\right)}{N}+\frac{\pi_{21} \left(1-\pi_{21}\right)}{N}+\frac{2\pi_{12}\pi_{21}}{N}\\ &=\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\\ V \left(p_{1\bullet }-p_{\bullet 1}\right)&=V \left(p_{1\bullet }\right)+V \left(p_{\bullet 1}\right)-2\mathrm{Cov} \left(p_{1\bullet },p_{\bullet 1}\right)\tag{1} \end{align} いっぽう、 \begin{align} V \left(p_{1\bullet }-p_{\bullet 1}\right)&=V \left[ \left(p_{11}+p_{12}\right)- \left(p_{11}+p_{21}\right)\right]\\ &=V \left(p_{12}-p_{21}\right)\tag{2} \end{align} よって、式 $(1).(2)$ より、 \begin{align} \frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}&=\frac{\pi_{1\bullet } \left(1-\pi_{1\bullet }\right)}{N}+\frac{\pi_{\bullet 1} \left(1-\pi_{\bullet 1}\right)}{N}-2\mathrm{Cov} \left(p_{1\bullet },p_{\bullet 1}\right)\\ 2\mathrm{Cov} \left(p_{1\bullet },p_{\bullet 1}\right)&=\frac{\pi_{1\bullet } \left(1-\pi_{1\bullet }\right)}{N}+\frac{\pi_{\bullet 1} \left(1-\pi_{\bullet 1}\right)}{N}-\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\\ &=\frac{\pi_{1\bullet }-\pi_{1\bullet }^2}{N}+\frac{\pi_{\bullet 1}-\pi_{\bullet 1}^2}{N}-\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\\ &=\frac{ \left(\pi_{11}+\pi_{12}\right)- \left(\pi_{11}+\pi_{12}\right)^2}{N}+\frac{ \left(\pi_{11}+\pi_{21}\right)- \left(\pi_{11}+\pi_{21}\right)^2}{N}-\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}-\pi_{21}\right)^2}{N}\\ &=\frac{ \left(\pi_{11}+\pi_{12}\right)- \left(\pi_{11}^2+2\pi_{11}\pi_{12}+\pi_{12}^2\right)}{N}+\frac{ \left(\pi_{11}+\pi_{21}\right)- \left(\pi_{11}^2+2\pi_{11}\pi_{21}+\pi_{21}^2\right)}{N}-\frac{ \left(\pi_{12}+\pi_{21}\right)- \left(\pi_{12}^2-2\pi_{12}\pi_{21}+\pi_{21}^2\right)}{N} &=\frac{2\pi_{11}-2 \left\{\pi_{11} \left(\pi_{11}+\pi_{12}+\pi_{21}\right)+\pi_{12}\pi_{21}\right\}}{N}\\ &=\frac{2 \left[\pi_{11}-\pi_{11} \left(\pi_{11}+\pi_{12}+\pi_{21}\right)-\pi_{12}\pi_{21}\right]}{N}\\ &=\frac{2 \left[\pi_{11} \left\{1- \left(\pi_{11}+\pi_{12}+\pi_{21}\right)\right\}-\pi_{12}\pi_{21}\right]}{N} &=\frac{2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}\right)}{N}\\ \mathrm{Cov} \left(p_{1\bullet },p_{\bullet 1}\right)&=\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{N} \end{align} 標本比率ベクトルを \begin{align} \boldsymbol{p}= \left(\begin{matrix}p_{1\bullet }\\p_{\bullet 1}\\\end{matrix}\right) \end{align} 期待値ベクトルを \begin{align} \boldsymbol{\pi}= \left(\begin{matrix}\pi_{1\bullet }\\\pi_{\bullet 1}\\\end{matrix}\right) \end{align} 分散・共分散行列を \begin{align} \boldsymbol{\Omega}= \left[\begin{matrix}\frac{\pi_{1\bullet } \left(1-\pi_{1\bullet }\right)}{N}&\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{N}\\\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{N}&\frac{\pi_{\bullet 1} \left(1-\pi_{\bullet 1}\right)}{N}\\\end{matrix}\right] \end{align} として、 \begin{gather} G \left(\boldsymbol{\pi}\right)=\theta=\log{\frac{\pi_{1\bullet }}{\pi_{\bullet 1}}}\\ G \left(\boldsymbol{p}\right)=\hat{\theta}=\log{\frac{p_{1\bullet }}{p_{\bullet 1}}} \end{gather} と変数変換する。 多変量のデルタ法を用いて $G \left(\boldsymbol{p}\right)$ を期待値 $E \left(\boldsymbol{p}\right)=\boldsymbol{\pi}$ まわりでテイラー展開すると、偏導関数ベクトルは、 \begin{align} \boldsymbol{H} \left(\boldsymbol{\pi}\right)= \left(\begin{matrix}\frac{G \left(\boldsymbol{\pi}\right)}{\partial\pi_{1\bullet }}\\\frac{G \left(\boldsymbol{\pi}\right)}{\partial\pi_{\bullet 1}}\\\end{matrix}\right)= \left[\begin{matrix}\frac{1}{\pi_{1\bullet }}\\-\frac{1}{\pi_{\bullet 1}}\\\end{matrix}\right] \end{align} 多変量のデルタ法の期待値と分散の公式より、 \begin{align} E \left[G \left(\boldsymbol{p}\right)\right]\cong G \left(\boldsymbol{\pi}\right)=\log{\frac{\pi_{1\bullet }}{\pi_{\bullet 1}}} \end{align} \begin{align} V \left[G \left(\boldsymbol{p}\right)\right]&=\frac{1}{N} \left[\begin{matrix}\frac{1}{\pi_{1\bullet }}&-\frac{1}{\pi_{\bullet 1}}\\\end{matrix}\right] \left[\begin{matrix}\pi_{1\bullet } \left(1-\pi_{1\bullet }\right)&\pi_{11}\pi_{22}-\pi_{12}\pi_{21}\\\pi_{11}\pi_{22}-\pi_{12}\pi_{21}&\pi_{\bullet 1} \left(1-\pi_{\bullet 1}\right)\\\end{matrix}\right] \left[\begin{matrix}\frac{1}{\pi_{1\bullet }}\\-\frac{1}{\pi_{\bullet 1}}\\\end{matrix}\right]\\ &=\frac{1}{N} \left[\begin{matrix}\frac{1}{\pi_{1\bullet }}&-\frac{1}{\pi_{\bullet 1}}\\\end{matrix}\right] \left[\begin{matrix}1-\pi_{1\bullet }-\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{\bullet 1}}\\\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{1\bullet }}- \left(1-\pi_{\bullet 1}\right)\\\end{matrix}\right]\\ &=\frac{1}{N} \left[\frac{1}{\pi_{1\bullet }} \left\{ \left(1-\pi_{1\bullet }\right)-\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{\bullet 1}}\right\}-\frac{1}{\pi_{\bullet 1}} \left\{\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{1\bullet }}- \left(1-\pi_{\bullet 1}\right)\right\}\right]\\ &=\frac{1}{N} \left(\frac{1-\pi_{1\bullet }}{\pi_{1\bullet }}-\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{1\bullet }\pi_{\bullet 1}}-\frac{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}}{\pi_{1\bullet }\pi_{\bullet 1}}+\frac{1-\pi_{\bullet 1}}{\pi_{\bullet 1}}\right)\\ &=\frac{\pi_{\bullet 1} \left(1-\pi_{1\bullet }\right)-2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}\right)+\pi_{1\bullet } \left(1-\pi_{\bullet 1}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{ \left(\pi_{\bullet 1}+\pi_{1\bullet }-2\pi_{1\bullet }\pi_{\bullet 1}\right)-2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{\bullet 1}+\pi_{1\bullet }-2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}+\pi_{1\bullet }\pi_{\bullet 1}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{11}+\pi_{12}+\pi_{11}+\pi_{21}-2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}+\pi_{1\bullet }\pi_{\bullet 1}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}-2 \left(\pi_{11}\pi_{22}-\pi_{12}\pi_{21}+\pi_{1\bullet }\pi_{\bullet 1}-\pi_{11}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}-2 \left\{\pi_{11}\pi_{22}-\pi_{12}\pi_{21}+ \left(\pi_{11}+\pi_{12}\right) \left(\pi_{11}+\pi_{21}\right)-\pi_{11}\right\}}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}-2 \left\{\pi_{11}\pi_{22}+\pi_{11}^2+\pi_{11} \left(\pi_{12}+\pi_{21}\right)-\pi_{11}\right\}}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}-2 \left\{\pi_{11} \left(\pi_{11}+\pi_{12}+\pi_{21}+\pi_{22}\right)-\pi_{11}\right\}}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}-2 \left(\pi_{11} \cdot 1-\pi_{11}\right)}{N\pi_{1\bullet }\pi_{1\bullet }}\\ &=\frac{\pi_{12}+\pi_{21}}{N\pi_{1\bullet }\pi_{1\bullet }} \end{align} 母比率 $\pi_i$ を一致推定量である標本比率 $p_i$ で置き換えると、漸近分散の一致推定量は、 \begin{align} {\hat{\sigma}}_1^2=\frac{p_{12}+p_{21}}{Np_{1\bullet }p_{\bullet 1}}=\frac{M}{Nn_{1\bullet }n_{\bullet 1}} \end{align} $\blacksquare$