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
$$
\begin{eqnarray}
y_i&=&\alpha+\beta x_i+\epsilon_i\;(i=1,\cdots,n)
\\\left\{\epsilon_i|i=1,\cdots,n\right\}&:&\epsilon_i \overset{iid}{\sim} N(0,\sigma^2)
\\&&\;\cdots\;独立同一分布(independent\;and\;identically\;distributed;\;IID,\;i.i.d.,\;iid)
\\&&\;\cdots\;\mathrm{E}\left[\epsilon_i\right]=0,\;\mathrm{V}\left[\epsilon_i\right]=\sigma^2,互いに独立\left(\mathrm{Cov}[\epsilon_i, \epsilon_j]=\left\{\begin{array}\;\mathrm{V}\left[\epsilon_i\right]=\sigma^2&(i=j)\\0&(i \neq j)\end{array}\right.\right)
\\\mathrm{V}\left[y_i\right]
&=&\mathrm{V}\left[\alpha+\beta x_i+\epsilon_i\right]
\\&=&\mathrm{V}\left[\epsilon_i\right]
\;\cdots\;\mathrm{V}\left[X\pm t\right]=\mathrm{V}\left[X\right]\;(t:分散をとることについて定数)
\\&=&\sigma^2
\\\mathrm{Cov}\left[y_i, y_j\right]
&=&\mathrm{E}\left[\left(y_i-\mathrm{E}\left[y_i\right]\right)\left(y_j-\mathrm{E}\left[y_j\right]\right)\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[X,Y\right]=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)\left(Y-\mathrm{E}\left[Y\right]\right)\right]}
\\&=&\mathrm{E}\left[\left(\alpha+\beta x_i+\epsilon_i-\mathrm{E}\left[\alpha+\beta x_i+\epsilon_i\right]\right)\left(\alpha+\beta x_j+\epsilon_j-\mathrm{E}\left[\alpha+\beta x_j+\epsilon_j\right]\right)\right]
\;\cdots\;y_i=\alpha+\beta x_i+\epsilon_i
\\&=&\mathrm{E}\left[\left(\alpha+\beta x_i+\epsilon_i-\alpha-\beta x_i-\mathrm{E}\left[\epsilon_i\right]\right)\left(\alpha+\beta x_j+\epsilon_j-\alpha-\beta x_j-\mathrm{E}\left[\epsilon_j\right]\right)\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[X\pm t\right]=\mathrm{E}\left[X\right]\pm t}
\\&=&\mathrm{E}\left[\left(\epsilon_i-\mathrm{E}\left[\epsilon_i\right]\right)\left(\epsilon_j-\mathrm{E}\left[\epsilon_j\right]\right)\right]
\\&=&\mathrm{Cov}\left[\epsilon_i, \epsilon_j\right]
\end{eqnarray}
$$
上記を踏まえて\((x_i-\bar{x})\)を加えた分散・共分散について
$$
\begin{eqnarray}
\mathrm{Cov}\left[(x_i-\bar{x})(y_i-\bar{y}), (x_j-\bar{x})(y_j-\bar{y})\right]
&=&(x_i-\bar{x})(x_j-\bar{x})\mathrm{Cov}\left[(y_i-\bar{y}), (y_j-\bar{y})\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[c_0X_i, c_1X_j\right]=c_0c_1\mathrm{Cov}\left[X_i,X_j\right]}
\\&=&(x_i-\bar{x})(x_j-\bar{x})\mathrm{E}\left[\left\{(y_i-\bar{y})-\mathrm{E}\left[y_i-\bar{y}\right]\right\}\left\{(y_j-\bar{y})-\mathrm{E}\left[y_j-\bar{y}\right]\right\}\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[X,Y\right]=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)\left(Y-\mathrm{E}\left[Y\right]\right)\right]}
\\&=&(x_i-\bar{x})(x_j-\bar{x})\mathrm{E}\left[\left\{y_i-\bar{y}-\mathrm{E}\left[y_i\right]+\bar{y}\right\}\left\{y_j-\bar{y}-\mathrm{E}\left[y_j\right]+\bar{y}\right\}\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[X\pm t\right]=\mathrm{E}\left[X\right]\pm t}
\\&=&(x_i-\bar{x})(x_j-\bar{x})\mathrm{E}\left[\left(y_i-\mathrm{E}\left[\bar{y}\right]\right)\left(y_j-\mathrm{E}\left[\bar{y}\right]\right)\right]
\\&=&(x_i-\bar{x})(x_j-\bar{x})\mathrm{Cov}\left[y_i, y_j\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[X,Y\right]=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)\left(Y-\mathrm{E}\left[Y\right]\right)\right]}
\\&=&\left\{\begin{array}
\;(x_i-\bar{x})^2\sigma^2&(i=j)
\\(x_i-\bar{x})(x_j-\bar{x})0&(i \neq j)
\end{array}\right.
\;\cdots\;\mathrm{Cov}\left[y_i,y_j\right]=\mathrm{Cov}\left[\epsilon_i,\epsilon_j\right]=\left\{\begin{array}\;\mathrm{V}\left[\epsilon_i\right]=\sigma^2&(i=j)\\0&(i \neq j)\end{array}\right.
\\
\mathrm{V}\left[(x_i-\bar{x})y_i\right]
&=&(x_i-\bar{x})^2\mathrm{V}\left[y_i\right]\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]}
\\&=&(x_i-\bar{x})^2\sigma^2\;\cdots\;上記i=jのケース
\end{eqnarray}
$$
\(S_{xy}\)の分散
$$
\begin{eqnarray}
\mathrm{V}\left[S_{xy}\right]
&=&\mathrm{V}\left[\sum_{i=1}^{n}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)\right]
\;\cdots\;S_{xy}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)
\\&=&
\sum_{i=1}^{n}\mathrm{V}\left[\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)\right]
+2\sum_{i\lt j}\mathrm{Cov}\left[\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right), \left(x_j-\bar{x}\right)\left(y_j-\bar{y}\right)\right]
\\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{V}\left[\sum_{i=1}^{n}X_i\right]
=\sum_{i=1}^{n}\mathrm{V}\left[X_i\right]+2\sum_{i\lt j}\mathrm{Cov}\left[X_i, X_j\right]}
\\&=&\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2\sigma^2+2\sum_{i\lt j}0
\\&&\;\cdots\;\mathrm{V}\left[\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)\right]=\left(x_i-\bar{x}\right)^2\sigma^2
\\&&\;\cdots\;\mathrm{Cov}\left[\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right), \left(x_j-\bar{x}\right)\left(y_j-\bar{y}\right)\right]=0\;(i\neq j)
\\&=&\sigma^2\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2
\;\cdots\;\sum_{i=0}^n cX_i=c\sum_{i=0}^n X_i
\\&=&\sigma^2S_{xx}
\;\cdots\;S_{xx}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2
\end{eqnarray}
$$
最小2乗推定量\(\hat{\beta}\)の分散
$$
\begin{eqnarray}
\mathrm{V}\left[\hat{\beta}\right]
&=&\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\right]
\;\cdots\;\hat{\beta}=\frac{S_{xy}}{S_{xx}},\;S_{xx}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2,\;S_{xy}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)
\\&=&\frac{1}{S_{xx}^2}\mathrm{V}\left[S_{xy}\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]}
\\&=&\frac{1}{S_{xx}^2}\sigma^2S_{xx}
\;\cdots\;\mathrm{V}\left[S_{xy}\right]=\sigma^2S_{xx}
\\&=&\frac{1}{S_{xx}}\sigma^2
\end{eqnarray}
$$
最小2乗推定量\(\hat{\alpha}\)の分散
$$
\begin{eqnarray}
\mathrm{V}\left[\hat{\alpha}\right]
&=&\mathrm{V}\left[\bar{y}-\hat{\beta}\bar{x}\right]
\;\cdots\;\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}
\\&=&\mathrm{V}\left[\bar{y}-\frac{S_{xy}}{S_{xx}}\bar{x}\right]
\;\cdots\;\hat{\beta}=\frac{S_{xy}}{S_{xx}},\;S_{xx}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2,\;S_{xy}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)
\\&=&\mathrm{V}\left[\bar{y}\right]+\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\bar{x}\right]
-2\mathrm{Cov}\left[\bar{y}, \frac{S_{xy}}{S_{xx}}\bar{x}\right]
\\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[X\pm Y\right]=\mathrm{V}\left[X\right]\pm 2\mathrm{Cov}\left[X,Y\right]+\mathrm{V}\left[Y\right]}
\\&=&\mathrm{V}\left[\bar{y}\right]+\mathrm{V}\left[\frac{S_{xy}}{S_{xx}}\bar{x}\right]-2\cdot0
\;\cdots\;\mathrm{Cov}\left[\bar{y}, \frac{S_{xy}}{S_{xx}}\bar{x}\right]=0\;(後述)
\\&=&\mathrm{V}\left[\bar{y}\right]
+\frac{\bar{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{\bar{x}^2}{S_{xx}^2}\sigma^2S_{xx}
\;\cdots\;\mathrm{V}\left[S_{xy}\right]=\sigma^2S_{xx}
\\&=&\frac{1}{n^2}\mathrm{V}\left[\sum_{i=1}^{n}y_i\right]
+\frac{\bar{x}^2}{S_{xx}}\sigma^2
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[cX\right]=c^2\mathrm{V}\left[X\right]}
\\&=&\frac{1}{n^2}
\left\{
\sum_{i=1}^{n}\mathrm{V}\left[y_i\right]
+2\sum_{i\lt j}\mathrm{Cov}\left[y_i, y_j\right]
\right\}
+\frac{\bar{x}^2}{S_{xx}}\sigma^2
\\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{V}\left[\sum_{i=1}^{n}X_i\right]
=\sum_{i=1}^{n}\mathrm{V}\left[X_i\right]+2\sum_{i\lt j}\mathrm{Cov}\left[X_i, X_j\right]}
\\&=&\frac{1}{n^2}\left\{
\sum_{i=1}^{n}\sigma^2
+2\sum_{i\lt j}0
\right\}
+\frac{\bar{x}^2}{S_{xx}}\sigma^2
\;\cdots\;\mathrm{V}\left[y_i\right]=\sigma^2,\;\mathrm{Cov}\left[y_i, y_j\right]=0
\\&=&\frac{1}{n^2}n\sigma^2
+\frac{\bar{x}^2}{S_{xx}}\sigma^2
\;\cdots\;\sum_{i=0}^n c=nc
\\&=&\left(\frac{1}{n}+\frac{\bar{x}^2}{S_{xx}}\right)\sigma^2
\end{eqnarray}
$$
最小2乗推定量\(\hat{\alpha}\)と\(\hat{\beta}\)の共分散
$$
\begin{eqnarray}
\mathrm{Cov}\left[\hat{\alpha},\hat{\beta}\right]
&=&\mathrm{E}\left[\left(\hat{\alpha}-\mathrm{E}\left[\hat{\alpha}\right]\right)\left(\hat{\beta}-\mathrm{E}\left[\hat{\beta}\right]\right)\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[X,Y\right]=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)\left(Y-\mathrm{E}\left[Y\right]\right)\right]}
\\&=&\mathrm{E}\left[
\left(
\left(\bar{y}-\hat{\beta}\bar{x}\right)
-\mathrm{E}\left[\bar{y}-\hat{\beta}\bar{x}\right]
\right)
\left(
\hat{\beta}-\mathrm{E}\left[\hat{\beta}\right]
\right)
\right]
\;\cdots\;\alpha=\bar{y}-\hat{\beta}\bar{x}
\\&=&\mathrm{E}\left[
\left(
\bar{y}-\frac{S_{xy}}{S_{xx}}\bar{x}
-\mathrm{E}\left[
\bar{y}-\frac{S_{xy}}{S_{xx}}\bar{x}
\right]
\right)
\left(
\frac{S_{xy}}{S_{xx}}
-\mathrm{E}\left[
\frac{S_{xy}}{S_{xx}}
\right]
\right)
\right]
\;\cdots\;\hat{\beta}=\frac{S_{xy}}{S_{xx}},\;S_{xx}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2,\;S_{xy}=\sum_{i=1}^{n}\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)
\\&=&\mathrm{E}\left[
\left(
\bar{y}-\frac{S_{xy}}{S_{xx}}\bar{x}
-\mathrm{E}\left[
\bar{y}
\right]
+\mathrm{E}\left[
\frac{S_{xy}}{S_{xx}}\bar{x}
\right]
\right)
\left(
\frac{S_{xy}}{S_{xx}}
-\mathrm{E}\left[
\frac{S_{xy}}{S_{xx}}
\right]
\right)
\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[X\pm Y\right]=\mathrm{E}\left[X\right]\pm\mathrm{E}\left[Y\right]}
\\&=&\mathrm{E}\left[
\left(
\bar{y}-\frac{S_{xy}}{S_{xx}}\bar{x}
-\mathrm{E}\left[
\bar{y}
\right]
+\frac{\bar{x}}{S_{xx}}\mathrm{E}\left[
S_{xy}
\right]
\right)
\left(
\frac{S_{xy}}{S_{xx}}
-\frac{1}{S_{xx}}\mathrm{E}\left[
S_{xy}
\right]
\right)
\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]}
\\&=&\mathrm{E}\left[
\left\{
\bar{y}-\mathrm{E}\left[\bar{y}\right]
-\frac{\bar{x}}{S_{xx}}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)
\right\}
\left\{
\frac{1}{S_{xx}}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)
\right\}
\right]
\\&=&\mathrm{E}\left[
-\frac{\bar{x}}{S_{xx}}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)
\frac{1}{S_{xx}}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)
\right]
\;\cdots\;\bar{y}-\mathrm{E}\left[\bar{y}\right]=\bar{y}-\bar{y}=0
\\&=&\mathrm{E}\left[
-\frac{\bar{x}}{S_{xx}^2}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)^2
\right]
\\&=&-\frac{\bar{x}}{S_{xx}^2}\mathrm{E}\left[
\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)^2
\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]}
\\&=&-\frac{\bar{x}}{S_{xx}^2}\sigma^2S_{xx}
\;\cdots\;\mathrm{E}\left[\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)^2\right]=\mathrm{V}\left[S_{xy}\right]=\sigma^2S_{xx}
\\&=&-\frac{\bar{x}}{S_{xx}}\sigma^2
\end{eqnarray}
$$
\(\mathrm{Cov}\left[\bar{y}, \frac{S_{xy}}{S_{xx}}\bar{x}\right]=0\)について
$$
\begin{eqnarray}
\mathrm{Cov}\left[\bar{y}, \frac{S_{xy}}{S_{xx}}\bar{x}\right]
&=&\mathrm{E}\left[\left(\bar{y}-\mathrm{E}\left[\bar{y}\right]\right)\left(\frac{S_{xy}}{S_{xx}}\bar{x}-\mathrm{E}\left[\frac{S_{xy}}{S_{xx}}\bar{x}\right]\right)\right]
\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/covariance.html}{\mathrm{Cov}\left[X,Y\right]=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)\left(Y-\mathrm{E}\left[Y\right]\right)\right]}
\\&=&\mathrm{E}\left[\left(\bar{y}-\bar{y}\right)\left(\frac{S_{xy}}{S_{xx}}\bar{x}-\frac{\bar{x}}{S_{xx}}\mathrm{E}\left[S_{xy}\right]\right)\right]
\;\cdots\;\mathrm{E}\left[\bar{y}\right]=\bar{y},\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-expected-value.html}{\mathrm{E}\left[cX\right]=c\mathrm{E}\left[X\right]}
\\&=&\mathrm{E}\left[0\cdot\frac{\bar{x}}{S_{xx}}\left(S_{xy}-\mathrm{E}\left[S_{xy}\right]\right)\right]
\\&=&\mathrm{E}\left[0\right]
\\&=&0
\end{eqnarray}
$$
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