Let $\hat{\theta}_{n}$ be an estimator $\theta$ obtained from $X_{i} \stackrel{i i d}{\sim} f(x ; \theta), i=1, \cdots, n .$ (a) Prove the following identity: $$ E(\hat{\theta- \theta)^{2}=\operatorname{Var}\left(\hat{\theta}_{n} i ght)+\left\ {\operatorname{Bias}\left(\hat{\theta}_{n} ight) ight 1}^{2} $$ where $\operatorname{Bias}\left(\hat{\theta}_{n} ight)=E\lef t(\hat{\theta}_{n} ight)-\theta$. (b) Show that an unbiased estimator $\hat{\theta}_{n}$ of $\theta$ is consistent if $\lim _{n ightarrow \infty} \operatorname[Var}\left(\hat{\theta}_{n} ight)=0$. SP.PC.064