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

Let ${\hat{\theta }}_n$ be the MLE and let $I(\theta )$ be the one-observation Fisher information matrix. One standard form of the Wald statistic isAny consistent information estimator gives the same asymptotics. Under $H_0$,
$W_n({\theta }_0)\Rightarrow {\chi }_p^2$. If
$c_{p,\alpha }$ is the $(1-\alpha )$ quantile of ${\chi }_p^2$, reject whenThe asymptotic type-I error is $\alpha$.