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It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than the fourth describe any useful properties of a distribution?

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Not an answer but one thing to keep in mind is that higher order moments require a lot more observations to get the first sig-fig. – isomorphismes Sep 17 '12 at 3:41
up vote 9 down vote accepted

Aside from special properties of a few numbers (e.g., 2), the only real reason to single out integer moments as opposed to fractional moments is convenience.

Higher moments can be used to understand tail behavior. For example, a centered random variable $X$ with variance 1 has subgaussian tails (i.e. $\mathbb{P}(|X| > t) < C e^{-ct^2}$ for some constants $c,C > 0$) if and only if $\mathbb{E}|X|^p \le (A \sqrt{p})^p$ for every $p\ge 1$ and some constant $A > 0$.

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the result you state for [sub]gaussian tails does not look right. according to the bound [$A\sqrt{p}$] you cite, the $p^{th}$ norm of a centered gaussian variable would not [in the limit] exceed 1. but the $p^{th}$ norm of a rv tends to its ess sup, which is $+\infty$ for a gaussian variable. – ronaf Sep 26 '10 at 2:00
Thanks for catching that. I forgot the exponent on the RHS; it's corrected now. – Mark Meckes Sep 27 '10 at 16:52
could you provide a reference for this result? – Gary Mar 21 '11 at 10:01
@Gary: unfortunately I don't know a (published or online) reference; it's part of the folklore of my field, spelled out in courses but written off as "simple and well-known" in papers. The proof is easy, though. Given the tail estimate, the moment estimate follows from integration by parts (i.e. $\mathbb{E} |X|^p = \int_0^\infty p t^{p-1} \mathbb{P}(|X| > t) dt$) and Stirling's formula. Given the moment estimate, the tail estimate follows by applying Markov's inequality and optimizing over $p$. – Mark Meckes Mar 24 '11 at 18:33

I get suspicious when I hear people ask about third and fourth moments. There are two common errors people often have in mind when they bring up the topic. I'm not saying that you are necessarily making these mistakes, but they do come up often.

First, it sounds like they implicitly believe that distributions can be boiled down to four numbers; they suspect that just two numbers is not enough, but three or four should be plenty.

Second, it sounds like hearkening back to the moment-matching approach to statistics that has largely lost out to maximum likelihood methods in contemporary statistics.

Update: I expanded this answer into a blog post.

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One example of use (interpretation is a better qualifier) of a higher moment: the fifth moment of a univariate distribution measures the asymmetry of its tails.

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But doesn't the third (central) moment do this in a more stable and practical way? – whuber Sep 20 '10 at 21:16
@Whuber:> the third is measuring overall asymmetry, which is not the same thing as tail asymmetry. Because of the higher exponent, the value of the fifth is nearly entirely determined by the tails. – user603 Sep 20 '10 at 22:31
@Kwak: Thank you for clarifying your meaning. Of course, the same response could be applied to any odd moment: they measure asymmetry further and further out in the tails. – whuber Sep 23 '10 at 22:07
@Whuber:> Of course. Note that even for a fair tailed distribution like the gaussian, by the 7th moment you are already in effect comparing the max to the min. – user603 Sep 23 '10 at 23:12
@Kwak: Two quick follow-up questions; no need to respond if you don't want. (1) "Fair tailed"?? (2) What are the min and max of a Gaussian? – whuber Sep 24 '10 at 13:29

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