Choosing the null hypothesis $p>p_0$ instead of $p\geq p_0$

Question

I have a very basic question, as I am a novice regarding statistics. Given the problem that I want to model, it would make sense to formulate the null hypothesis of my one-sided test as $H_0: p>p_0$ instead of $H_0: p\geq p_0$. Is there a reason why all the modelling I see in books is done by taking the not strict inequality with respect to $p_0$ in the formulation of $H_0$?

Answer 1

It is just convention to include the “equals” part as part of the null hypothesis. The power function $\pi(\theta\vert \delta)$ might also be maximized on the boundary for the null hypothesis, which is related to the size of the test, so if there were strict inequality it might not be possible to have an exact value for the size of the test $\alpha(\delta)=\sup _{\theta\in \Omega_0}\pi(\theta\vert \delta)$ when testing the hypotheses

$$H_0: \theta\in\Omega_0\\ H_1: \theta\in\Omega_1$$

with $\Omega_0$ having the form $(-\infty, a)$.

But in terms of choosing a null and alternative hypothesis, typically they are formulated so that the type I error is most to be avoided. For instance, confirmation of one’s own theory, if it were false, is considered a more serious error in academics than falsely saying one’s (correct) theory is false. So since type I error is rejecting the null $H_0$ when it is true, the alternative hypothesis $H_1$ would be the theory in this case.