Generalised likelihood-ratio test
Section of Generalised likelihood-ratio tests, part of Statistical inference
Generalised likelihood-ratio test
| Type | Section |
| Taught in | Generalised likelihood-ratio tests |
| Part | Statistical inference |
Let L(\theta) be the likelihood and write
\Lambda=\frac{\sup_{\theta\in\Omega_0}L(\theta)}{\sup_{\theta\in\Omega_1}L(\theta)}.
The numerator is the largest likelihood allowed by the null. The denominator is the largest likelihood allowed by the alternative. When the null set sits inside the alternative set, the denominator is the unrestricted maximum and \Lambda\le 1. Small \Lambda means the null cannot reach the fit of the larger model. The generalised likelihood-ratio test rejects H_0 when \Lambda is smaller than a threshold chosen to control the type I error.
For a normal mean with known variance the supremum under H_0:\mu=\mu_0 is the likelihood at \mu_0, and the supremum under H_1:\mu\ne\mu_0 is the likelihood at \overline X. The ratio is a monotone function of |\overline X-\mu_0|/(\sigma/\sqrt n), so the test is the two-sided z test. Nothing new happens until a parameter remains free under the null.
The statistic is usually reported as
-2\log\Lambda=2\bigl\{\ell(\widehat\theta_1)-\ell(\widehat\theta_0)\bigr\},
the difference of maximised log-likelihoods, or as a difference of deviances. Deviance is this difference against a saturated model. A difference of deviances between two nested models cancels the saturated term and leaves -2\log\Lambda.