# Lecture 12. D) Wald Test

$T_{w}\left(X_{1}..X_{n}\right)=\frac{\left(\widehat{\theta}_{ML}-\theta_{0}\right)^{2}}{\left[-\frac{\partial^{2}}{\partial\theta^{2}}l\left(\left.\widehat{\theta}_{ML}\right|X_{1}..X_{n}\right)\right]^{-1}}$.
This test can also be motivated as an approximation to the LRT. The principle of the Wald test is to reject the null hypothesis when $\left|\widehat{\theta}_{ML}-\theta_{0}\right|$ is large.
The denominator is a measure of informativeness of $\widehat{\theta}_{ML}$. It increases the rejection rate as the uncertainty over $\widehat{\theta}_{ML}$ decreases. (If the estimator is more precise, then we reject for lower distances).