The book "Testing Statistical Hypotheses" by Lehman describes in section 4.5, a test for comparing the rates of two Poisson point-processes. This is the "uniformly most powerful" (UMP) test for this process. Now, a data set I was looking at had higher variance than mean which violates the assumption of the count of events in an interval being Poisson distributed (which has the same variance as mean). So, I decided that a Compound Poisson process might be a good fit for such data. What this means is that the arrivals are still governed by a Poisson process, but each time there is an arrival, another random variable governs the number of point-events (instead of one event per arrival). The distribution I considered for the distribution of the number of point events with each arrival was Binomial.
Now, I was expecting the power of the UMP Poisson test to be compromised if used on this Compound Binomial process. So, I ran some simulations generating first from a null hypothesis (two identical distributions) and then alternate (Poisson arrival rate increased by some effect size) to get the false positive and false negative rates ($\alpha$ and $\beta$ respectively). The $\alpha$-$\beta$ curve I thought would be "better" (less $\beta$ for a given $\alpha$) when I simulated from a Poisson than when I used a Compound Poisson. However, as the figure below shows, this didn't seem to be the case (note only the blue and orange curves which are on top of each other). It seems the power of the test stays the same, which is very surprising. Note that the x-axis on the first graph is the actual $\alpha$ you get from the test (which might be different from the $\alpha$ you compare the p-value to if your distributional assumption is violated).
The only consequence of using the UMP Poisson test on a Compound Poisson seems to be that the $\alpha$ you set differs from the $\alpha$ you get. The relationship is shown in the figure below. But apart from that, the power isn't compromised. Does anyone have some intuition or a proof as to why this might be?