0
$\begingroup$

According to Williams, D.: Weighing the Odds the p-value of observed data in the likelihood ratio setting is defined as

$$\mathrm{p_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big)$$

where

$$ \mathrm{lr}(y) := \frac{\mathrm{sup}_{\theta \in B_A} \mathrm{lhd}(\theta, y)}{\mathrm{sup}_{\theta \in B_0} \mathrm{lhd}(\theta, y)} $$

where $\mathrm{lhd}$ is the likelihood function. He further calls it a 'convoluted concept', which got me thinking what is the reason for this comment. One thing I realized, which I did not realize before, is that in the first equation the supremum can be achieved by a value of $\theta^*$ that could have never generated the $y^{obs}$ (could be $f_{\theta^*}(y^{obs}) = 0).$ So I thought maybe it would make sense to define something like

$$\mathrm{p^{mod}_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \bigg (\mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big) \mathbb{P}(y^{obs} \vert \theta) \bigg )$$

Has this been done before? Does it have some name? What about if we change it to

$$\mathrm{p^{mod 2}_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \bigg (\mathbb{P}\big(\mathrm{lr}(Y) \geq \mathrm{lr}(y^{obs}) \vert \theta \big) \mathbb{I}_{\mathbb{P}(y^{obs} \vert \theta) > 0} \bigg )$$

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.