How will p-values behave when fitting normal/Poisson to binomial?

Question

I know p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for say n = 1000. Will my p-value still be uniformly distributed or as binomial converges to normal(Poisson), p-values mostly will be in 0.8-1?

Answer 1

I think what you are referring to is sampling a binomial random variable $X$ and then looking at the distribution of $F(X)$ where $F$ is the binomial CDF. This is actually not uniform; it is a sort of discrete approximation of a uniform variable. In the case where normal approximation becomes valid, it converges to a uniform variable. In the case where Poisson approximation becomes valid, it does not converge to a uniform variable, but rather to the distribution of $F(P)$ where $P$ has the limiting Poisson distribution and $F$ is the limiting Poisson CDF. Again this isn't uniform.