We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of inter-renewal process is $p(x)$ or $q(x)$.
We want to do this hypothesis testing in such a way that the sum of error probabilities (type I error and type II error) is minimum. What is the sum of error probabilities in this case?
I have solved this problem for the case that $p(x)$ and $q(x)$ are two exponential distributions so the renewal processes are Poisson. Can you please help to solve it for the general case?