Past exam of the options II course of the University of Oxford / 2010 / O / O4 / a / Solution

The maximum-likelihood estimator is a maximizerWriting ${\ell }_n=\log L_n$ and the score $S_n={\nabla }_{\theta }{\ell }_n$, the expected Fisher information matrix iswhere the equality uses the usual regularity conditions. For independent identically distributed observations, ${\mathcal{I}}_n=n{\mathcal{I}}_1$.

Under the standard identifiability, interior-parameter, differentiability and nonsingular-information assumptions, the large-sample distribution isor approximately $\hat{\theta }\sim N(\theta ,{\mathcal{I}}_n(\theta {)}^{-1})$. Boundary parameters need not obey this regular approximation.