Simple null
Theorem: If regularity conditions (A)-(D) hold, then
\[(\bth-\bts) \Tr \fI_n(\bts) (\bth-\bts) \inD \chi_d^2.\]This follows directly from the asymptotic normality of the MLE.
Nuisance parameters
Theorem: If regularity conditions (A)-(D) hold and $\bt_0=\bts_1$ (i.e., if $H_0$ is true), then
\[\sqrt{n}(\bth_1-\bt_0) \inD \Norm(\zero, \fV_{11}),\]where $\fV_{11}^{-1} = \fI_{11} - \fI_{12} \fI_{22}^{-1} \fI_{21}$ is the portion of $\fI(\bts)$ corresponding to $\bt_1$.
This follows directly from the asymptotic normality of the MLE and the marginal distribution of a multivariate normal.
Confidence intervals
If our parameter of interest is a scalar, then we have simple closed-form expressions for confidence intervals:
\[\th_j \pm z_{1-\alpha/2} \sqrt{\mathcal{V}^n_{jj}(\bth)}.\]