統計学実践ワークブックの行間埋め　第8章

\begin{eqnarray} E_{\theta}[(\hat{\theta}-\theta)^2] &=& (E[(\hat{\theta}-\theta)])^2+V[(\hat{\theta}-\theta)] \\ \\ &=& (E[\hat{\theta}]-\theta)^2+V[\hat{\theta}] \;\;\;&...V[X+c]=V[X]のため\\ \\ &=& (b_{\theta}(\hat{\theta}))^2+V[\hat{\theta}] \\ \end{eqnarray}

\begin{eqnarray} \frac{\partial}{\partial \beta}\displaystyle \sum_{i=1}^n (y_i-\hat{\beta} x_i)^2 &=& -2\displaystyle \sum_{i=1}^n x_i(y_i-\hat{\beta} x_i) \\ \\ &=& -2 (\displaystyle \sum_{i=1}^n x_iy_i-\sum_{i=1}^n\hat{\beta} x_i^2)\\ \\ &=0 \\ \\ &\Leftrightarrow& \displaystyle \sum_{i=1}^n\hat{\beta} x_i^2=\displaystyle \sum_{i=1}^n x_iy_i \\ \\ &\Leftrightarrow& \hat{\beta}=\frac{\displaystyle \sum_{i=1}^n x_iy_i}{\displaystyle \sum_{i=1}^nx_i^2} \\ \end{eqnarray}

こちらの解説などを参考

こちらの解説などを参考