Rao-Cramer Lower Bound

Definition

Let $X_1,X_2,\dots ,X_n$ be Random Sample with PDF $f(x|\theta )$ with the R0 ~ R4 regularity conditions and $Y=u(X_1,X_2,\dots ,X_n)$ be a Statistic with $E(Y)=k(\theta )$, then $\mathrm{Var}(Y)\ge \frac{k^{\prime }(\theta {)}^2}{nI(\theta )}$

Facts

Let $Y=u(X_1,X_2,\dots ,X_n)$ be an unbiased estimator of $\theta$, then $E(Y)=k(\theta )=\theta$. Thus, by a Rao-Cramer Lower Bound, $\mathrm{Var}(Y)\ge \frac{1}{nI(\theta )}$