Theory and Methods of Statistics

7.1.2 Asymptotic Efficiency of MLE

What is the connection between the information bound $1/I(\theta )=1/{\text{E}}_{\theta }[{\dot{l}}^2(X)]$ for the variance of an unbiased estimator of θ and the fact that the MLE of ${\widehat{\theta }}_n$ of θ has the asymptotic distribution $\sqrt{n}({\widehat{\theta }}_n-\theta )\stackrel{\mathcal{L}}{\to }N(0,1/I(\theta ))?$ [For simplicity, we are considering the case of a 1-dim parameter θ.]

We have seen that

(i) under regularity conditions on the pdf/pmf and restrictions on estimators, nVar_θ[T_n] ≥ 1/I(θ) for all unbiased ...