Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 29L (Principles of Statistics)

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 $\alpha$.

With ${\ell }_n(\theta )={\sum }_i\log f(X_i,\theta )$, define the likelihood-ratio statisticWilks' theorem gives ${\Lambda }_n\Rightarrow {\chi }_p^2$ under $H_0$, so reject whenThis again has asymptotic type-I error $\alpha$.

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