Rao-Blackwell Theorem

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\theta )$, where $\theta \in \Omega$, $Y_1=u_1(X_1,X_2,\dots ,X_n)$ be a Sufficient Statistic for $\theta$, $Y_2=u_2(X_1,X_2,\dots ,X_n)$ be a Unbiased Estimator for $\theta$. And define $\phi (Y_1):=E(Y_2|Y_1)$, then $\mathrm{Var}[\phi (Y_1)]\le \mathrm{Var}(Y_2)\wedge E(\phi (Y_1))=\theta$ $\phi (Y_1)$ is an Unbiased Estimator, and it has a lower variance than $Y_2$