Wald Tests and Score Tests

The generalized likelihood ratio test has one practical drawback that Wilks' theorem does not fix: computing $\Lambda(x)$ requires fitting two models — the restricted MLE $\hat\theta_0$ and the unrestricted MLE $\hat\theta$ — even though the derivation note showed both fits converge to the same quadratic-form limit in the end. This note builds two large-sample tests that each need only one of the two fits. The Wald test uses only the unrestricted MLE $\hat\theta$, asking how far it sits from $\theta_0$ in information-weighted units — exactly the statistic already used to build the Wald confidence interval. The score (Rao) test uses only the restricted MLE $\hat\theta_0$, asking how steeply the log-likelihood is still climbing at the null value — and, remarkably, never needs to fit the unrestricted model at all. All three tests share the same asymptotic $\chi^2(k)$ null distribution by Remark 1 of the derivation note; the choice between them, in practice, comes down to which model is cheaper to fit and how close the true parameter sits to a boundary of $\Theta$.

Learning objectives. By the end of this note you should be able to: