不偏推定量の分散(平均二乗誤差)

推定したパラメタ(不偏推定量)の分散(平均二乗誤差)

$$ \begin{eqnarray} \mathrm{E}\left[\left(\hat{\theta}-\theta\right)^2\right] &=&\mathrm{E}\left[ \hat{\theta}^2-2\hat{\theta}\theta+\theta^2 \right]\\ &=&\mathrm{E}\left[\hat{\theta}^2\right] -2\theta\mathrm{E}\left[\hat{\theta}\right] +\theta^2\mathrm{E}\left[1\right]\\ &=&\mathrm{E}\left[\hat{\theta}^2\right] -2\theta\mathrm{E}\left[\hat{\theta}\right] +\theta^2 +\mathrm{E}\left[\hat{\theta}\right]^2 -\mathrm{E}\left[\hat{\theta}\right]^2 \;\cdots\;\mathrm{E}\left[\hat{\theta}\right]^2 -\mathrm{E}\left[\hat{\theta}\right]^2=0\\ &=&\mathrm{E}\left[\hat{\theta}\right]^2 -2\theta\mathrm{E}\left[\hat{\theta}\right] +\theta^2 +\mathrm{E}\left[\hat{\theta}^2\right] -\mathrm{E}\left[\hat{\theta}\right]^2\\ &=&\left(\mathrm{E}\left[\hat{\theta}\right]-\theta\right)^2 +\mathrm{V}\left[\hat{\theta}\right] \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/discrete-random-variable-variance.html}{\mathrm{V}\left[X\right]=\mathrm{E}\left[X^2\right]-\mathrm{E}\left[X\right]^2}\\ &=&\left(\theta-\theta\right)^2 +\mathrm{V}\left[\hat{\theta}\right] \;\cdots\;\hat{\theta}は不偏推定量なので\mathrm{E}\left[\hat{\theta}\right]=\theta\\ &=&\mathrm{V}\left[\hat{\theta}\right]\\ \end{eqnarray} $$