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