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For any probability distribution function (PDF), $p(x)$, which has finite moments $\left<X^k\right>$ defined upto $k=N$, is it possible to say something about the heavyness of the tails by considering the ratio of the moments $\left<X^m\right>/\left<X^n\right>$ where $(n<m\le N)$?

Further, can we also quantify this and define an estimator of tail heavyness, $E_H$, based on the ratio of moments?

For example, for two distribution $p_1(x)$ and $p_2(y)$, can we say that $ E_{H_1} < E_{H_2} $, if the ratio of the moments follow the relationship, $\frac{\left<X^m\right>}{\left<X^n\right>} < \frac{\left<Y^m\right>}{\left<Y^n\right>}$ for any $m>n$.

user35952
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    What exactly do you mean by "heavyness"? According to most standard definitions of "heavy" or "long" or "fat" tails, these ratios don't tell you much at all. Far more relevant would be the supremum (among positive real numbers) of the $k$ for which $\langle X^k\rangle$ is finite. For estimating such a property, moments are particularly bad because unless the tail is the very opposite of "heavy" (in any sense), the higher moment estimates are extremely variable. – whuber Jun 28 '23 at 13:34
  • I developed a similar measure of tail weight (leverage) here , https://stats.stackexchange.com/a/481022/102879, one that has a very precise interpretation of the type that you want. Yes, it looks like you have hit upon a reasonable approach. You might consider absolute and also centered moments, though. One advantage of moment-based definitions of tail weight (unlike those involving asymptotes of densities) is that they directly apply to actual data, with the same precise interpretation as indicated in my post. – BigBendRegion Jul 25 '23 at 15:27

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