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For GLRT, the ratio is:

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

but we instead use:

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

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

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

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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.

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  • $\begingroup$ Thanks for the comment! I'm wondering why we use the union for an unconstrained MLE instead of the MLE for w1 $\endgroup$ Commented May 21, 2023 at 16:37
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    $\begingroup$ @user6132211 : I wanted a function that is equal to $1$ when there is no evidence against the null hypothesis and less than $1$ otherwise. $\endgroup$ Commented May 23, 2023 at 18:05

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