Simple null

Theorem: If regularity conditions (A)-(D) hold, then

(\hat{θ} - θ^{*})^{⊤} I_{n} (θ^{*}) (\hat{θ} - θ^{*}) \overset{d}{⟶} χ_{d}^{2} .

This follows directly from the asymptotic normality of the MLE.

Nuisance parameters

Theorem: If regularity conditions (A)-(D) hold and $θ_{0} = θ_{1}^{*}$ (i.e., if $H_{0}$ is true), then

\sqrt{n} ({\hat{θ}}_{1} - θ_{0}) \overset{d}{⟶} N (0, V_{11}),

where $V_{11}^{- 1} = I_{11} - I_{12} I_{22}^{- 1} I_{21}$ is the portion of $I (θ^{*})$ corresponding to $θ_{1}$ .

If our parameter of interest is a scalar, then we have simple closed-form expressions for confidence intervals:

{\hat{θ}}_{j} \pm z_{1 - α / 2} \sqrt{V_{j j}^{n} (\hat{θ})} .