Is there a definition of p-value for interval null hypothesis?

Question

For testing a simple null hypothesis $\theta=\theta_0$ vs alternative $\theta\ne\theta_0$, the definition of p-value given a UMP (usually) test $\mathbb{1}(T>k)$ (reject when $T>k$ where $k$ be chosen such that $E(\mathbb{1}(T>k))=\alpha$ for some statistics $T$ and realized value $t_0$ is $$P_{\theta_0}(T>t_0)$$

My question is is there a definition of p-value for testing hypothesis such as $H_0:\theta\in(\theta_1,\theta_2)$ vs $H_1:\theta\notin(\theta_1,\theta_2)$. Furthermore, if the test is not in the form $\mathbb{1}(T>k)$, what shall we do?

Answer 1

If I remember my stats class correctly, it's the maximum of the p values for all the simple hypotheses in the composite hypothesis. This is why if you do a one-sided $t$ test for the null hypothesis $\theta\le 0$, you do the calculation assuming $\theta=0$, because this gives you the largest probability of exceeding the observed $t$ statistic.

Googling "p value composite hypothesis" seems to back up my recollection. For example, see page 3 of these notes.

Answer 2

Why do you think that the same approach does not work here? It is just a matter of defining the test statistics suitably, but to be honest that is not what people really do in practice.

In practice, what most people do is to base the test on whether a $1-\alpha$ confidence interval for $\theta$ lies fully within $H_1$ (=does not include any parameter values from $H_0$). To get a p-value one would typically look at two one-sided tests at level $\alpha/2$. In the manner you have your hypotheses, it would be one of $H_{01}: \theta \geq \theta_1$ versus $H_{A1}: \theta < \theta_1$ and the other of $H_{02}: \theta \leq \theta_2$ versus $H_{A2}: \theta > \theta_2$ (for these you know how to derive the p-values). Then you take $p = 2 \times \min(p_1, p_2)$ as your p-value for this particular testing problem.

By the way, are you sure you got your hypotheses the right way around? I have mostly seen this kind of situation with null and alternative swapped around.