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I'm trying to approximate an unknown distribution by a truncated Edgeworth series, with cumulants/central moments estimated from a large sample.

I notice though that I am getting negative tail probability densities. What would be the restrictions on my estimates of the sample central moments to guarantee positive density? Is there an accessible paper or set of lecture notes treating this problem?

Frido
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    Gram-Charlier and Edgeworth expansions have an undesirable tendency to sometimes produce small negative frequencies, particularly in the tails. It is a consequence of taking an approximation with less than an infinite number of terms. In an ideal world, these negatives frequencies could be avoided by taking higher order expansions. This in turn requires higher order moments, which in turn have high variance and may be unreliable unless the sample size is sufficiently large. Trying to force positive density sounds inconsistent with the essence of the approximation (while still integrating to 1) – wolfies Mar 18 '23 at 12:44
  • @wolfies I agree that philosophically there is some inconsistency, but nevertheless I am faced with the problem of having to estimate a density from a sample, and this density needs to be positive and integrate to 1. Gram-Charlier / Edgeworth or saddlepoint method seems to be the most straightforward way to proceed. Except that for GC/Edgeworth there is this issue with possilble negative densities when the series is truncated. I'm not sure if the saddlepoint approximation suffers from the same potential isse. – Frido Mar 18 '23 at 13:11
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    There are always other methods. Have you tried Pearson fitting (just requires first 4 sample moments) or Johnson method, or Non-parametric kernel density estimation? If you can post your data, or the sample central moments and sample mean, different methods can be compared. – wolfies Mar 18 '23 at 13:31
  • @wolfies I just read a related post and a link to a chapter in your book (https://stats.stackexchange.com/a/95234/256959).Tthanks for suggesting Pearson, I'll try that – Frido Mar 18 '23 at 16:36

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