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、 $$\begin{array}{c}e\sim \mathrm{B}(N,{\pi }_{11})\\ f\sim \mathrm{B}(N,{\pi }_{12})\\ g\sim \mathrm{B}(N,{\pi }_{21})\\ e+f=n_{1\bullet }\sim \mathrm{B}(N,{\pi }_{1\bullet })\\ e+g=n_{\bullet 1}\sim \mathrm{B}(N,{\pi }_{\bullet 1})\end{array}$$ 二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_{11}\sim \mathrm{N}[{\pi }_{11},\frac{{\pi }_{11}(1-{\pi }_{11})}{N}]\\ p_{12}\sim \mathrm{N}[{\pi }_{12},\frac{{\pi }_{12}(1-{\pi }_{12})}{N}]\\ p_{21}\sim \mathrm{N}[{\pi }_{21},\frac{{\pi }_{21}(1-{\pi }_{21})}{N}]\\ p_{1\bullet }\sim \mathrm{N}[{\pi }_{1\bullet },\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}]\\ p_{\bullet 1}\sim \mathrm{N}[{\pi }_{\bullet 1},\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}]\end{array}$$ ここで、和の分散の公式より、 $$\begin{array}{rl}V(p_{12}-p_{21})& =V\left(p_{12}\right)+V\left(p_{21}\right)-2\mathrm{Cov}(p_{12},p_{21})\\ & =\frac{{\pi }_{12}(1-{\pi }_{12})}{N}+\frac{{\pi }_{21}(1-{\pi }_{21})}{N}+\frac{2{\pi }_{12}{\pi }_{21}}{N}\\ & =\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ \text{(1)}& V(p_{1\bullet }-p_{\bullet 1})& =V\left(p_{1\bullet }\right)+V\left(p_{\bullet 1}\right)-2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})\end{array}$$ いっぽう、 $$\begin{array}{rl}V(p_{1\bullet }-p_{\bullet 1})& =V[(p_{11}+p_{12})-(p_{11}+p_{21})]\\ \text{(2)}& & =V(p_{12}-p_{21})\end{array}$$ よって、式 $(1).(2)$ より、 $$\begin{array}{rl}\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}& =\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}+\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}-2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})\\ 2\mathrm{Cov}(p_{1\bullet },p_{\bullet 1})& =\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}+\frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{{\pi }_{1\bullet }-{\pi }_{1\bullet }^2}{N}+\frac{{\pi }_{\bullet 1}-{\pi }_{\bullet 1}^2}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{({\pi }_{11}+{\pi }_{12})-{({\pi }_{11}+{\pi }_{12})}^2}{N}+\frac{({\pi }_{11}+{\pi }_{21})-{({\pi }_{11}+{\pi }_{21})}^2}{N}-\frac{({\pi }_{12}+{\pi }_{21})-{({\pi }_{12}-{\pi }_{21})}^2}{N}\\ & =\frac{({\pi }_{11}+{\pi }_{12})-({\pi }_{11}^2+2{\pi }_{11}{\pi }_{12}+{\pi }_{12}^2)}{N}+\frac{({\pi }_{11}+{\pi }_{21})-({\pi }_{11}^2+2{\pi }_{11}{\pi }_{21}+{\pi }_{21}^2)}{N}-\frac{({\pi }_{12}+{\pi }_{21})-({\pi }_{12}^2-2{\pi }_{12}{\pi }_{21}+{\pi }_{21}^2)}{N}& =\frac{2{\pi }_{11}-2\{{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21})+{\pi }_{12}{\pi }_{21}\}}{N}\\ & =\frac{2[{\pi }_{11}-{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21})-{\pi }_{12}{\pi }_{21}]}{N}\\ & =\frac{2[{\pi }_{11}\{1-({\pi }_{11}+{\pi }_{12}+{\pi }_{21})\}-{\pi }_{12}{\pi }_{21}]}{N}& =\frac{2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})}{N}\\ \mathrm{Cov}(p_{1\bullet },p_{\bullet 1})& =\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}\end{array}$$ 標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_{1\bullet }\\ p_{\bullet 1}\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_{1\bullet }\\ {\pi }_{\bullet 1}\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_{1\bullet }(1-{\pi }_{1\bullet })}{N}& \frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}\\ \frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{N}& \frac{{\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})}{N}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\log \frac{{\pi }_{1\bullet }}{{\pi }_{\bullet 1}}\\ G\left(\mathit{p}\right)=\hat{\theta }=\log \frac{p_{1\bullet }}{p_{\bullet 1}}\end{array}$$ と変数変換する。 多変量のデルタ法を用いて $G\left(\mathit{p}\right)$ を期待値 $E\left(\mathit{p}\right)=\mathit{\pi }$ まわりでテイラー展開すると、偏導関数ベクトルは、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_{1\bullet }}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_{\bullet 1}}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_{1\bullet }}\\ -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\log \frac{{\pi }_{1\bullet }}{{\pi }_{\bullet 1}}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\frac{1}{N}\left[\begin{array}{cc}\frac{1}{{\pi }_{1\bullet }}& -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\left[\begin{array}{cc}{\pi }_{1\bullet }(1-{\pi }_{1\bullet })& {\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}\\ {\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}& {\pi }_{\bullet 1}(1-{\pi }_{\bullet 1})\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_{1\bullet }}\\ -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\\ & =\frac{1}{N}\left[\begin{array}{cc}\frac{1}{{\pi }_{1\bullet }}& -\frac{1}{{\pi }_{\bullet 1}}\end{array}\right]\left[\begin{array}{c}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 }}-(1-{\pi }_{\bullet 1})\end{array}\right]\\ & =\frac{1}{N}[\frac{1}{{\pi }_{1\bullet }}\{(1-{\pi }_{1\bullet })-\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{\bullet 1}}\}-\frac{1}{{\pi }_{\bullet 1}}\{\frac{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}}{{\pi }_{1\bullet }}-(1-{\pi }_{\bullet 1})\}]\\ & =\frac{1}{N}(\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}})\\ & =\frac{{\pi }_{\bullet 1}(1-{\pi }_{1\bullet })-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})+{\pi }_{1\bullet }(1-{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{({\pi }_{\bullet 1}+{\pi }_{1\bullet }-2{\pi }_{1\bullet }{\pi }_{\bullet 1})-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{\bullet 1}+{\pi }_{1\bullet }-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{11}+{\pi }_{12}+{\pi }_{11}+{\pi }_{21}-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2({\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+{\pi }_{1\bullet }{\pi }_{\bullet 1}-{\pi }_{11})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}{\pi }_{22}-{\pi }_{12}{\pi }_{21}+({\pi }_{11}+{\pi }_{12})({\pi }_{11}+{\pi }_{21})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}{\pi }_{22}+{\pi }_{11}^2+{\pi }_{11}({\pi }_{12}+{\pi }_{21})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2\{{\pi }_{11}({\pi }_{11}+{\pi }_{12}+{\pi }_{21}+{\pi }_{22})-{\pi }_{11}\}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}-2({\pi }_{11}\cdot 1-{\pi }_{11})}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\\ & =\frac{{\pi }_{12}+{\pi }_{21}}{N{\pi }_{1\bullet }{\pi }_{1\bullet }}\end{array}$$ 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、 $$\begin{array}{r}{\hat{\sigma }}_1^2=\frac{p_{12}+p_{21}}{N{p}_{1\bullet }{p}_{\bullet 1}}=\frac{M}{N{n}_{1\bullet }{n}_{\bullet 1}}\end{array}$$ $◼$