Wald test

Asymptotic hypothesis test for testing a simple null against a composite alternative: $H_0:\theta ={\theta }_0$ vs $H_1:\theta \ne {\theta }_0$. The test is based on any estimator of ${\theta }_0$ that, under the null, obeys a CLT of the form

$$T(\hat{\theta })=\frac{\hat{\theta }-{\theta }_0}{\sqrt{\mathbb{V}(\hat{\theta })}}\to N(0,1),$$

(We can also replace the denominator with any consistent estimator of $\sqrt{\mathbb{V}(\hat{\theta })}$ ). Thus, we treat the statistic $T(\hat{\theta })$ as a normal under the null and reject if $|T(\hat{\theta })|\ge z_{\alpha /2}$.