単回帰における最小二乗推定量の分散(variance)・共分散(covariance)

単回帰における最小二乗推定量$\hat{\alpha },\hat{\beta }$の分散(variance)・共分散(covariance)

単回帰における観測値$y_i$の分散・共分散について

$$\begin{array}{rcl}{y}_i& =& \alpha +\beta x_i+{ϵ}_i(i=1,\cdots ,n)\\ \{{ϵ}_i|i=1,\cdots ,n\}& :& {ϵ}_i\stackrel{iid}{\sim }N(0,{\sigma }^2)\\ & & \cdots \mathrm{独}\mathrm{立}\mathrm{同}\mathrm{一}\mathrm{分}\mathrm{布}(independentandidenticallydistributed;IID,i.i.d.,iid)\\ & & \cdots \mathrm{E}\left[{ϵ}_i\right]=0,\mathrm{V}\left[{ϵ}_i\right]={\sigma }^2,\mathrm{互}\mathrm{い}\mathrm{に}\mathrm{独}\mathrm{立}(\mathrm{Cov}[{ϵ}_i,{ϵ}_j]=\{\begin{array}{}\mathrm{V}\left[{ϵ}_i\right]={\sigma }^2& (i=j)\\ 0& (i\ne j)\end{array})\\ \mathrm{V}\left[y_i\right]& =& \mathrm{V}[\alpha +\beta x_i+{ϵ}_i]\\ & =& \mathrm{V}\left[{ϵ}_i\right]\cdots \mathrm{V}[X\pm t]=\mathrm{V}\left[X\right](t:\mathrm{分}\mathrm{散}\mathrm{を}\mathrm{と}\mathrm{る}\mathrm{こ}\mathrm{と}\mathrm{に}\mathrm{つ}\mathrm{い}\mathrm{て}\mathrm{定}\mathrm{数})\\ & =& {\sigma }^2\\ \mathrm{Cov}[y_i,y_j]& =& \mathrm{E}\left[(y_i-\mathrm{E}\left[y_i\right])(y_j-\mathrm{E}\left[y_j\right])\right]\cdots \mathrm{Cov}[X,Y]=\mathrm{E}\left[(X-\mathrm{E}\left[X\right])(Y-\mathrm{E}\left[Y\right])\right]\\ & =& \mathrm{E}\left[(\alpha +\beta x_i+{ϵ}_i-\mathrm{E}[\alpha +\beta x_i+{ϵ}_i])(\alpha +\beta x_j+{ϵ}_j-\mathrm{E}[\alpha +\beta x_j+{ϵ}_j])\right]\cdots y_i=\alpha +\beta x_i+{ϵ}_i\\ & =& \mathrm{E}\left[(\alpha +\beta x_i+{ϵ}_i-\alpha -\beta x_i-\mathrm{E}\left[{ϵ}_i\right])(\alpha +\beta x_j+{ϵ}_j-\alpha -\beta x_j-\mathrm{E}\left[{ϵ}_j\right])\right]\cdots \mathrm{E}[X\pm t]=\mathrm{E}\left[X\right]\pm t\\ & =& \mathrm{E}\left[({ϵ}_i-\mathrm{E}\left[{ϵ}_i\right])({ϵ}_j-\mathrm{E}\left[{ϵ}_j\right])\right]\\ & =& \mathrm{Cov}[{ϵ}_i,{ϵ}_j]\end{array}$$ 上記を踏まえて$(x_i-\overline{x})$を加えた分散・共分散について $$\begin{array}{rcl}\mathrm{Cov}[(x_i-\overline{x})(y_i-\overline{y}),(x_j-\overline{x})(y_j-\overline{y})]& =& (x_i-\overline{x})(x_j-\overline{x})\mathrm{Cov}[(y_i-\overline{y}),(y_j-\overline{y})]\cdots \mathrm{Cov}[c_0{X}_i,c_1{X}_j]=c_0{c}_1\mathrm{Cov}[X_i,X_j]\\ & =& (x_i-\overline{x})(x_j-\overline{x})\mathrm{E}\left[\{(y_i-\overline{y})-\mathrm{E}[y_i-\overline{y}]\}\{(y_j-\overline{y})-\mathrm{E}[y_j-\overline{y}]\}\right]\cdots \mathrm{Cov}[X,Y]=\mathrm{E}\left[(X-\mathrm{E}\left[X\right])(Y-\mathrm{E}\left[Y\right])\right]\\ & =& (x_i-\overline{x})(x_j-\overline{x})\mathrm{E}\left[\{y_i-\overline{y}-\mathrm{E}\left[y_i\right]+\overline{y}\}\{y_j-\overline{y}-\mathrm{E}\left[y_j\right]+\overline{y}\}\right]\cdots \mathrm{E}[X\pm t]=\mathrm{E}\left[X\right]\pm t\\ & =& (x_i-\overline{x})(x_j-\overline{x})\mathrm{E}\left[(y_i-\mathrm{E}\left[\overline{y}\right])(y_j-\mathrm{E}\left[\overline{y}\right])\right]\\ & =& (x_i-\overline{x})(x_j-\overline{x})\mathrm{Cov}[y_i,y_j]\cdots \mathrm{Cov}[X,Y]=\mathrm{E}\left[(X-\mathrm{E}\left[X\right])(Y-\mathrm{E}\left[Y\right])\right]\\ & =& \{\begin{array}{}(x_i-\overline{x}{)}^2{\sigma }^2& (i=j)\\ (x_i-\overline{x})(x_j-\overline{x})0& (i\ne j)\end{array}\cdots \mathrm{Cov}[y_i,y_j]=\mathrm{Cov}[{ϵ}_i,{ϵ}_j]=\{\begin{array}{}\mathrm{V}\left[{ϵ}_i\right]={\sigma }^2& (i=j)\\ 0& (i\ne j)\end{array}\\ \mathrm{V}[(x_i-\overline{x})y_i]& =& (x_i-\overline{x}{)}^2\mathrm{V}\left[y_i\right]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& (x_i-\overline{x}{)}^2{\sigma }^2\cdots \mathrm{上}\mathrm{記}i=j\mathrm{の}\mathrm{ケ}\mathrm{ー}\mathrm{ス}\end{array}$$

$S_{xy}$の分散

$$\begin{array}{rcl}\mathrm{V}\left[S_{xy}\right]& =& \mathrm{V}\left[\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\right]\cdots S_{xy}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \sum _{i=1}^n\mathrm{V}\left[(x_i-\overline{x})(y_i-\overline{y})\right]+2\sum _{i<j}\mathrm{Cov}[(x_i-\overline{x})(y_i-\overline{y}),(x_j-\overline{x})(y_j-\overline{y})]\\ & & \cdots \mathrm{V}\left[\sum _{i=1}^n{X}_i\right]=\sum _{i=1}^n\mathrm{V}\left[X_i\right]+2\sum _{i<j}\mathrm{Cov}[X_i,X_j]\\ & =& \sum _{i=1}^n{(x_i-\overline{x})}^2{\sigma }^2+2\sum _{i<j}0\\ & & \cdots \mathrm{V}\left[(x_i-\overline{x})(y_i-\overline{y})\right]={(x_i-\overline{x})}^2{\sigma }^2\\ & & \cdots \mathrm{Cov}[(x_i-\overline{x})(y_i-\overline{y}),(x_j-\overline{x})(y_j-\overline{y})]=0(i\ne j)\\ & =& {\sigma }^2\sum _{i=1}^n{(x_i-\overline{x})}^2\cdots \sum _{i=0}^{n}c{X}_i=c\sum _{i=0}^n{X}_i\\ & =& {\sigma }^2{S}_{xx}\cdots S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2\end{array}$$

最小2乗推定量$\hat{\beta }$の分散

$$\begin{array}{rcl}\mathrm{V}\left[\hat{\beta }\right]& =& \mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\right]\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}},S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2,S_{xy}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \frac{1}{S_{xx}^2}\mathrm{V}\left[S_{xy}\right]\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& \frac{1}{S_{xx}^2}{\sigma }^2{S}_{xx}\cdots \mathrm{V}\left[S_{xy}\right]={\sigma }^2{S}_{xx}\\ & =& \frac{1}{S_{xx}}{\sigma }^2\end{array}$$

最小2乗推定量$\hat{\alpha }$の分散

$$\begin{array}{rcl}\mathrm{V}\left[\hat{\alpha }\right]& =& \mathrm{V}[\overline{y}-\hat{\beta }\overline{x}]\cdots \hat{\alpha }=\overline{y}-\hat{\beta }\overline{x}\\ & =& \mathrm{V}[\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x}]\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}},S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2,S_{xy}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \mathrm{V}\left[\overline{y}\right]+\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\overline{x}\right]-2\mathrm{Cov}[\overline{y},\frac{S_{xy}}{S_{xx}}\overline{x}]\\ & & \cdots \mathrm{V}[X\pm Y]=\mathrm{V}\left[X\right]\pm 2\mathrm{Cov}[X,Y]+\mathrm{V}\left[Y\right]\\ & =& \mathrm{V}\left[\overline{y}\right]+\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\overline{x}\right]-2\cdot 0\cdots \mathrm{Cov}[\overline{y},\frac{S_{xy}}{S_{xx}}\overline{x}]=0(\mathrm{後}\mathrm{述})\\ & =& \mathrm{V}\left[\overline{y}\right]+\frac{{\overline{x}}^2}{S_{xx}^2}\mathrm{V}\left[S_{xy}\right]\\ & =& \mathrm{V}\left[\frac{1}{n}\sum _{i=1}^n{y}_i\right]+\frac{{\overline{x}}^2}{S_{xx}^2}{\sigma }^2{S}_{xx}\cdots \mathrm{V}\left[S_{xy}\right]={\sigma }^2{S}_{xx}\\ & =& \frac{1}{n^2}\mathrm{V}\left[\sum _{i=1}^n{y}_i\right]+\frac{{\overline{x}}^2}{S_{xx}}{\sigma }^2\cdots \mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]\\ & =& \frac{1}{n^2}\{\sum _{i=1}^n\mathrm{V}\left[y_i\right]+2\sum _{i<j}\mathrm{Cov}[y_i,y_j]\}+\frac{{\overline{x}}^2}{S_{xx}}{\sigma }^2\\ & & \cdots \mathrm{V}\left[\sum _{i=1}^n{X}_i\right]=\sum _{i=1}^n\mathrm{V}\left[X_i\right]+2\sum _{i<j}\mathrm{Cov}[X_i,X_j]\\ & =& \frac{1}{n^2}\{\sum _{i=1}^n{\sigma }^2+2\sum _{i<j}0\}+\frac{{\overline{x}}^2}{S_{xx}}{\sigma }^2\cdots \mathrm{V}\left[y_i\right]={\sigma }^2,\mathrm{Cov}[y_i,y_j]=0\\ & =& \frac{1}{n^2}n{\sigma }^2+\frac{{\overline{x}}^2}{S_{xx}}{\sigma }^2\cdots \sum _{i=0}^{n}c=nc\\ & =& (\frac{1}{n}+\frac{{\overline{x}}^2}{S_{xx}}){\sigma }^2\end{array}$$

最小2乗推定量$\hat{\alpha }$と$\hat{\beta }$の共分散

$$\begin{array}{rcl}\mathrm{Cov}[\hat{\alpha },\hat{\beta }]& =& \mathrm{E}\left[(\hat{\alpha }-\mathrm{E}\left[\hat{\alpha }\right])(\hat{\beta }-\mathrm{E}\left[\hat{\beta }\right])\right]\cdots \mathrm{Cov}[X,Y]=\mathrm{E}\left[(X-\mathrm{E}\left[X\right])(Y-\mathrm{E}\left[Y\right])\right]\\ & =& \mathrm{E}\left[((\overline{y}-\hat{\beta }\overline{x})-\mathrm{E}[\overline{y}-\hat{\beta }\overline{x}])(\hat{\beta }-\mathrm{E}\left[\hat{\beta }\right])\right]\cdots \alpha =\overline{y}-\hat{\beta }\overline{x}\\ & =& \mathrm{E}\left[(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x}-\mathrm{E}[\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x}])(\frac{S_{xy}}{S_{xx}}-\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\right])\right]\cdots \hat{\beta }=\frac{S_{xy}}{S_{xx}},S_{xx}=\sum _{i=1}^n{(x_i-\overline{x})}^2,S_{xy}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})\\ & =& \mathrm{E}\left[(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x}-\mathrm{E}\left[\overline{y}\right]+\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\overline{x}\right])(\frac{S_{xy}}{S_{xx}}-\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\right])\right]\cdots \mathrm{E}[X\pm Y]=\mathrm{E}\left[X\right]\pm \mathrm{E}\left[Y\right]\\ & =& \mathrm{E}\left[(\overline{y}-\frac{S_{xy}}{S_{xx}}\overline{x}-\mathrm{E}\left[\overline{y}\right]+\frac{\overline{x}}{S_{xx}}\mathrm{E}\left[S_{xy}\right])(\frac{S_{xy}}{S_{xx}}-\frac{1}{S_{xx}}\mathrm{E}\left[S_{xy}\right])\right]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& \mathrm{E}\left[\{\overline{y}-\mathrm{E}\left[\overline{y}\right]-\frac{\overline{x}}{S_{xx}}(S_{xy}-\mathrm{E}\left[S_{xy}\right])\}\left\{\frac{1}{S_{xx}}(S_{xy}-\mathrm{E}\left[S_{xy}\right])\right\}\right]\\ & =& \mathrm{E}[-\frac{\overline{x}}{S_{xx}}(S_{xy}-\mathrm{E}\left[S_{xy}\right])\frac{1}{S_{xx}}(S_{xy}-\mathrm{E}\left[S_{xy}\right])]\cdots \overline{y}-\mathrm{E}\left[\overline{y}\right]=\overline{y}-\overline{y}=0\\ & =& \mathrm{E}[-\frac{\overline{x}}{S_{xx}^2}{(S_{xy}-\mathrm{E}\left[S_{xy}\right])}^2]\\ & =& -\frac{\overline{x}}{S_{xx}^2}\mathrm{E}\left[{(S_{xy}-\mathrm{E}\left[S_{xy}\right])}^2\right]\cdots \mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& -\frac{\overline{x}}{S_{xx}^2}{\sigma }^2{S}_{xx}\cdots \mathrm{E}\left[{(S_{xy}-\mathrm{E}\left[S_{xy}\right])}^2\right]=\mathrm{V}\left[S_{xy}\right]={\sigma }^2{S}_{xx}\\ & =& -\frac{\overline{x}}{S_{xx}}{\sigma }^2\end{array}$$

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$\mathrm{Cov}[\overline{y},\frac{S_{xy}}{S_{xx}}\overline{x}]=0$について

$$\begin{array}{rcl}\mathrm{Cov}[\overline{y},\frac{S_{xy}}{S_{xx}}\overline{x}]& =& \mathrm{E}\left[(\overline{y}-\mathrm{E}\left[\overline{y}\right])(\frac{S_{xy}}{S_{xx}}\overline{x}-\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\overline{x}\right])\right]\cdots \mathrm{Cov}[X,Y]=\mathrm{E}\left[(X-\mathrm{E}\left[X\right])(Y-\mathrm{E}\left[Y\right])\right]\\ & =& \mathrm{E}\left[(\overline{y}-\overline{y})(\frac{S_{xy}}{S_{xx}}\overline{x}-\frac{\overline{x}}{S_{xx}}\mathrm{E}\left[S_{xy}\right])\right]\cdots \mathrm{E}\left[\overline{y}\right]=\overline{y},\mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]\\ & =& \mathrm{E}[0\cdot \frac{\overline{x}}{S_{xx}}(S_{xy}-\mathrm{E}\left[S_{xy}\right])]\\ & =& \mathrm{E}\left[0\right]\\ & =& 0\end{array}$$