Most asymptotic results in statistics prove that as n \rightarrow \infty an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood function. I believe there’s a similar result in Bayesian literature, the “Bayesian Central Limit Theorem”, which shows that the posterior converges asymptotically to a normal as n \rightarrow \infty
My question is – does the distribution converge to something “before” it becomes normal, based on the third term in the Taylor series? Or is this not possible to do in general?
You are searching for the Edgeworth series aren’t you?
(note that Edgeworth died in 1926, should be in most famous statisticians? )