Past exam of the options I course of the University of Oxford / 2011 / O / 1 / a / i

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 $I_n(\theta )=n{I}_1(\theta )$.