Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 28J / a / Solution

For observations $x=(x_1,\dots ,x_n)$, the likelihood function isA maximum-likelihood estimator is any measurable choiceFor one observation in a regular scalar model, the score function isand the Fisher information iswhere the second identity requires the usual differentiation-under-the-integral regularity. For $n$ independent observations, Tensorization of Fisher information gives $I_n(\theta )=nI(\theta )$.