Some Implications

If one has access to [math]I\left(\theta_{0}\right)^{-1}[/math], or one can approximate it via some estimator say [math]\widehat{I\left(\theta_{0}\right)}[/math] and/or [math]\widehat{I\left(\theta_{ML}\right)}[/math], one can use the normal distribution for hypothesis testing as long as [math]n[/math] is large.
The point above applies, even if one does not know the exact distribution of the test.
It can be shown that the LRT, Wald, and LM tests are asymptotically equivalent. The proof, perhaps unsurprisingly, uses Taylor expansion.

Lecture 15. B) Some Implications