Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 29K (Principles of Statistics)

Under the usual regularity conditions, consistency and asymptotic normality of a maximum likelihood estimator givewhere $I({\theta }_0)$ is the one-observation Fisher information matrix. In a scalar model this reads $N(0,I({\theta }_0{)}^{-1})$.

The multivariate Wald statistic for a candidate parameter $\theta \in {\mathbb{R}}^p$ isReplacing $I({\hat{\theta }}_n)$ by any consistent estimator of $I({\theta }_0)$ gives the same limit.

Under $P_{{\theta }_0}$, part (a) and consistency of the information matrix imply, by Slutsky theorem, thatConsequentlyusing the characterization of the chi-squared distribution as the squared norm of a standard normal vector.

If $q_{p,1-\alpha }$ is the $(1-\alpha )$ quantile of ${\chi }_p^2$, an asymptotic $(1-\alpha )$ confidence region is

LetThen the null hypothesis is $R{\theta }_0=r$. Define the Wald statistic for linear restrictions

Under $H_0$, part (a) and the continuous mapping theorem givewhereThe matrix $V_0$ is positive definite because $I({\theta }_0)$ is positive definite and $R$ has full row rank. Consistency givesTherefore Slutsky theorem yieldsand applying the continuous squared-norm map proves rigorously that

Let $q_{k,1-\alpha }$ be the $(1-\alpha )$ quantile of ${\chi }_k^2$. Rejecting $H_0$ whenhas rejection probability tending to $\alpha$ under every fixed parameter satisfying $H_0$, and is therefore an asymptotically valid level-$\alpha$ test.