Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 26K / c / Solution

A maximum-likelihood estimate ${\hat{\theta }}_n$ is a measurable maximizer of ${\ell }_n(\theta )={\sum }_i\log f(X_i,\theta )$ over $\Theta$. Under identifiability, consistency conditions, an interior true parameter, sufficient differentiability and finite positive Fisher information, the standard limits areThe variance in the normal distribution is the reciprocal of the per-observation Fisher information, not of $nI$. Strong consistency needs additional hypotheses if it is to replace convergence in probability; the regular asymptotic result stated here does not require claiming it.