Lehmann-Scheffe 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 complete sufficient Statistic for $\theta$, then If $E[\phi (Y_1)]=\theta$, then $\phi (Y_1)$ is the unique UMVUE of $\theta$

Facts

Let $Y_1$ be a complete sufficient Statistic for $\theta$, $Y_2$ be a Unbiased Estimator of $q(\theta )$

• $\exists h,E[h(Y_1)]=q(\theta )$, then $h(Y_1)$ is a unique UMVUE of $q(\theta )$
• By the Rao-Blackwell Theorem, $E(Y_2|Y_1)$ is a unique UMVUE of $q(\theta )$