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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

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