Let's say I have the following observations from many Bernoulli distributions with different p (p1, p2, ..):
Observations from Distribution 1: 10 successes, 100,000 trials, p_hat = 0.0001 Observations from Distribution 2: 0 successes, 100 trials, p_hat = 0 Observations from Distribution 3: 4 successes, 60,000 trials, p_hat = 0.00007
I want to order these distributions by their true probabilities of success and get rid of the ones that have low probability of success. However, because of the inherent nature of these distributions, the probability of success is so low, that if I use a standard Wald and Wilson confidence interval for Bernoulli distributions, the results don't make too much sense.
Is there a standard statistical way to deal with these types of problems? Or do I have to resort to some self defined heuristics to remove distributions with low probability of success?