Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 29L / c

For scalar $\theta$, Taylor's theorem about the MLE, where
${\ell }_n^{\prime }({\hat{\theta }}_n)=0$, gives some ${\tilde{\theta }}_n$ between ${\theta }_0$ and ${\hat{\theta }}_n$ such thatThe Wald statistic isUnder the null, consistency gives
${\hat{\theta }}_n,{\tilde{\theta }}_n\to {\theta }_0$ in probability. A uniform law of large numbers for the observed information on a neighborhood of ${\theta }_0$ givesCancellation of the common squared displacement and Slutsky's theorem prove the Wald and likelihood-ratio asymptotic equivalence