Re: normal distribution
[ QUOTE ]
The problem is with the expectation that such a theorem would be of value here. This theorem provides an upper bound for all distributions (satisfying the assumptions); in particular, it applies to "worst case" distributions. While I wouldn't call the distribution from one poker hand "very close" to normal (close is relative anyway...), it is certainly closer than some "worst case" examples we could dream up. So clearly this case would converge quicker.
[/ QUOTE ]
Absolutely. However, one thing to notice is that the "worst case" among distributions that share the same first 3 moments with the distribution of one poker hand is "nicer" than the "worst case" among those that share only the mean and SD. And, under mild assumptions, the more moments we know, the closer we get to describing the true shape of the distribution.
So, does anyone know, is there a Berry-Esseen-type theorem which utilizes the 4th moment? How about one which uses the first n moments? Something like this might be of use, provided we had raw data on these higher moments. In a case like that, the "worst case" would be much closer to the actual case, since we are specifying more moments and, hence, specifying more of the shape of the distribution.
(I feel I've been rambling and unclear. Hopefully, this makes sense.)
|