A Qualitative meaning of central moments higher than 4th order

[dgrosel]

The qualitative meaning of central moments up to 4th order (mean, variance, skewness, and kurtosis) is well documented and explained in literature. I am posting this question because I am interested to know if any of the higher moments (5th, 6th, and so on) has some clear qualitative meaning as well.

In particular, I am wondering if any of the high-order moments (or some combination of these) could be perhaps well-suited to detect a distribution that is a bit "bumpy" (compared for example, to a normal distribution) but possibly symmetric.

Thank you,
Daniel

 

[andrewkirk]

It's a good question - one I also have wondered about on occasion. It led me to the idea that we could investigate this by finding plotting the statistical distributions that have certain moments and seeing what they look like. For instance, comparing the distributions whose central moments are all zero except for the first five being (0,1,0,0,0) for one and (0,1,0,0,1) for the other would give an idea of what effect the fifth moment had. We could do the same comparison with distributions that had nonzero 3rd and/or 4th moments to see what interaction there was with those.

Like you I wondered whether some of the higher moments might generate bumps - local maxima and minima.

What's needed then is a recipe to derive the pdf of a distribution from its moments. That's where I ran out of enthusiasm. But perhaps there are better resources available for this now. This note looks like it gives two alternative recipes.

 

[mathman]

https://en.wikipedia.org/wiki/Moment-generating_function

If you have all the moments, the distribution function can be obtained from the characteristic function by inverse Fourier transform. Above reference describes the characteristic function in terms of the moments.

 

[chiro]

Hey dgrosel.

Building on what mathman said, the Fourier transform has a very good intuitive interpretation based on the spectrum interpretation of the transform itself.

It might also help you to think of looking at the moments as if they were components of orthogonal functions (which is studied in Harmonic Analysis).

Usually when you want to make sense of functions (the MGF is a function) you can use techniques in harmonic analysis to break the function down like it was a vector (like you do with basis elements and project the initial vector to the different basis vectors except that a function is now a vector in an infinite dimensional space).

One can do this with a specific distribution (like Normal) or in general and what it can show is the kind of information contributed when a particular "basis vector" is involved.

The theory of harmonic analysis is different when the interval is finite versus infinite and the infinite type is when Wavelets are involved.

It can be quite involved but it's the only general approach that I can think of that can be utilized for such a general question.

 

[Number Nine]

High-order moments (about the mean), in general, don't have clear "intuitive" interpretations in the way that variance and skewness do. In general, because higher moments involve increasing powers, they place greater and greater emphasis on the tails. Higher moments of odd-order, for example, measure how much of the asymmetry in the distribution is caused by differences in the tails, while higher moment sof even order measure how much of the spread is due to the tails.