Let \(T_n\) be an unbiased estimator of \(g(\theta)\) based on a random sample of size \(n\). Under certain regularity conditions, the Cramer-Rao lower bound gives \[ V_\theta[T_n]\ge\frac{(g'(\theta))^2}{nI(\theta)}, \] where \[ I(\theta)=E\left[\left( \frac{d\log f_{\theta}(x)}{d\theta}\right)^2\right]=-E\left[-\frac{d^2\log f_{\theta}(x)}{d\theta^2}\right] \] is the Fisher Information.
A sequence of estimators \(\{T_n,n\ge1\}\) is CAN estimators of \(g(\theta)\) if the asymptotic distribution of \(\sqrt{n}(T_n-g(\theta))\) is normal.
It was thought that when i.i.d observations were considered, the variance of the limiting distribution of \(\sqrt{n}(T_n-g(\theta))\) has the lower bound \(\frac{(g'(\theta))^2}{nI(\theta)}\)(CRLB).
An estimator \(T_n\) for which the stated lower bound is attained for the asymptotic distribution, is said to be efficient.
Let \(X_1,\ldots,X_n\stackrel{\text{iid}}\sim f_\theta(x)\). Let \(L_n(\theta)=\prod_{i=1}^nf_\theta(x_i)\) and let \(\hat\theta_n\) denote the MLE of \(\theta\). We know that \(\sqrt{n}(\hat \theta_n-\theta)\stackrel{d}\rightarrow N(0,I^{-1}(\theta))\)(Asymptotic normality of MLE). So, one might expect \(E[\hat\theta_n]=\theta+o_p(n^{-\frac{1}{2}})\). However, it is not true in general.