Simple null

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

u (θ^{*})^{⊤} I_{n}^{- 1} (θ^{*}) u (θ^{*}) \overset{d}{⟶} χ_{d}^{2} .

This follows directly from the distribution of the distribution of the score.

Note that the consistency of the MLE is not required here since the MLE is not involved.

Nuisance parameters

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

u_{1} ({\hat{θ}}_{0})^{⊤} V_{11}^{n} ({\hat{θ}}_{0}) u_{1} ({\hat{θ}}_{0}) \overset{d}{⟶} χ_{r}^{2},

where ${\hat{θ}}_{0}$ is the restricted MLE.

The proof of this theorem is a homework assignment.