Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / O / 1 / a

For the sample likelihood function $L(\theta )={\prod }_{i}f(X_i;\theta )$ and log-likelihood $\ell =\log L$, the expected Fisher information isThe second equality uses the usual regularity assumptions permitting differentiation under the integral and a parameter-independent support. Independent identically distributed observations give $I_n(\theta )=n{I}_1(\theta )$.

Under the standard consistency and regularity hypotheses for the maximum-likelihood estimator,Equivalently, for large samples $\hat{\theta }$ is approximately $N(\theta ,I_n(\theta {)}^{-1})$. In a standard-error estimate, the unknown parameter in the information may be replaced by the consistent estimate $\hat{\theta }$.