Past exam of the options I course of the University of Oxford / 2012 / O / O1 / a

Let ${\ell }_n(\theta )={\sum }_{i=1}^n\log f(X_i;\theta )$ be the log-likelihood. Under the usual regularity conditions, the expected Fisher information in the whole sample isThe equality of the two expressions uses mean-zero score and permitted differentiation of the normalization identity. Independent observations make the information additive. The maximum-likelihood estimator has asymptotic normality:The factor $n$ is included in $I_n$; it must not be inserted a second time into the variance.