Moments of a distribution - any use for partial or higher moments? 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?
 A: 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.
A: 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 $A > 0$.
A: 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.
