The reason of introducing the Berry-Esseen is that it proves such gap is inevitable in approximation (doesn’t matter whether a sampling distribution is Poisson, Gamma, binomial, etc) and it tells that there’s lower bound in gaps although case by case the gap sizes would be different let alone sample sizes. One would surprise how large n can be in order to make the gap small enough to a tolerable level. Note that it assumes iid and poses some constraints on moments. Theorems are there to be modified by relaxing conditions and assumptions, say from iid to independent, to prove other problems. Such tailoring yields a new theorem with another name or corollaries. Instead of saying >25 is a golden absolute rule and 10> is wrong in X-ray spectral grouping (counts in bins), I’d like to say more care is needed when data are binned/grouped. It’s hard to know those counts in each group satisfy these assumptions and produce small enough gap of tolerance for traditional the x-sigma error bar construction based on the delta chi method.

Fortunately, there are many efforts to minimize the impact of gaps in your plots with low count data while lessening sacrifice of innate information. They look so far just rules of thumb without mathematical justifications; at least documentations from XSPEC and Sherpa gave me such impression (some literatures take 20 intead of 25, suggest averaging errors of neighbors but do not provide definition/function/explanation to define degrees of neighborhood, etc). Quite room for mathematical statisticians to accompany astronomers’ lonely quest and to guide as well.

]]>