# Generalized Likelihood Ratio Test - Why is the denominator a union

For GLRT, the ratio is:

$$\Lambda^* = \frac{\max_{\theta \in \omega_0} L(\theta)}{\max_{\theta \in \omega_1}L(\theta)}$$

$$\Lambda = \frac{\max_{\theta \in \omega_0}L(\theta)}{\max_{\theta \in \omega_0 \cup \omega_1}L(\theta)}$$

I'm not understanding what the rationale behind this is and would appreciate any pointers.

• Isn't the second one $\Lambda$ the standard definition of likelihood ratio in hypotheses testing theory? Commented May 20, 2023 at 22:16
• I believe it's the first one for neyman pearson and uniformally most powerful. glrt uses the second one Commented May 21, 2023 at 16:38

I'm assuming you mean there is a null hypothesis stating that the parameter $$\theta$$ is in $$\omega_0$$ and $$\omega_1$$ is just the complement of $$\omega_0$$.
The unconstrained MLE for $$\theta$$ is $$\widehat{\theta\,} = \operatorname*{argmax}_{\theta\,\in\,\omega_0\, \cup\,\omega_1} L(\theta).$$
If $$\widehat{\theta\,} \in\omega_0$$ then, according to what you say "we instead use", the ratio
$$\Lambda = \frac{\max_{\theta\,\in\,\omega_0} L(\theta)}{\max_{\theta\,\in\,\omega_0\,\cup\,\omega_1} L(\theta)}$$
is equal to $$1.$$ Thus it is only when there is some evidence against the null hypothesis that $$\Lambda<1.$$ The stronger the evidence against the null hypothesis, the smaller $$\Lambda$$ is.
• @user6132211 : I wanted a function that is equal to $1$ when there is no evidence against the null hypothesis and less than $1$ otherwise. Commented May 23, 2023 at 18:05