= Solution
The <multivariate Wald statistic> for a candidate parameter $\theta\in\mathbb R^p$ is
$$
W_n(\theta)
=n(\widehat\theta_n-\theta)^T
I(\widehat\theta_n)
(\widehat\theta_n-\theta).
$$
Replacing $I(\widehat\theta_n)$ 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>, that
$$
I(\widehat\theta_n)^{1/2}
\sqrt n(\widehat\theta_n-\theta_0)
\xrightarrow d N_p(0,I_p).
$$
Consequently
$$
\boxed{W_n(\theta_0)\xrightarrow d\chi_p^2,}
$$
using 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
$$
\boxed{
C_n=\{\theta\in\mathbb R^p:W_n(\theta)\leq q_{p,1-\alpha}\}.
}
$$
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