Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 28K (Principles of Statistics)

For one observation with density $f(x,\theta )$, the Fisher information matrix at ${\theta }_0$ isHereSince ${\mathrm{Cov}}_{{\theta }_0}(X)=I_p$, the Fisher information of a multivariate normal location model isThe information in the full independent sample is $n{I}_p$ by Tensorization of Fisher information.

The maximum-likelihood estimator isUnder the score-outer-product convention stated in the question,This is the sample covariance matrix with divisor $n$. The strong law of large numbers, applied entrywise, givesThus the empirical score outer-product information consistently estimates the per-observation Fisher information.

The multivariate Wald statistic for a proposed value $\theta$ isBecause ${\hat{i}}_n\to I_p$ andSlutsky theorem gives

If $q_{p,1-\alpha }$ is the $(1-\alpha )$-quantile of the chi-squared distribution with $p$ degrees of freedom, an asymptotic $(1-\alpha )$ confidence region isIn this known-identity-covariance model one may replace ${\hat{i}}_n$ by $I_p$; then the confidence ball is exact. For an individual coordinate, the corresponding exact confidence interval is