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

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?

• I believe the theory is outlined in great generality and detail at stats.stackexchange.com/questions/31/…. Is there any aspect of your situation that you think is not adequately covered in that thread?
– whuber
Commented Apr 18, 2017 at 3:29

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.
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.