Lecture 12. D) Wald Test

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Wald Test

The Wald test statistic is given by

[math]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}}[/math].

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 [math]\left|\widehat{\theta}_{ML}-\theta_{0}\right|[/math] is large.

The denominator is a measure of informativeness of [math]\widehat{\theta}_{ML}[/math]. It increases the rejection rate as the uncertainty over [math]\widehat{\theta}_{ML}[/math] decreases. (If the estimator is more precise, then we reject for lower distances).