# Qualitative meaning of central moments higher than 4th order

• A
• dgrosel
In summary, the qualitative meaning of central moments up to 4th order is well-documented and explained in literature. While there is no clear intuitive interpretation for higher moments, they can be useful in detecting a distribution with "bumpy" characteristics. The distribution function can be obtained from the characteristic function by inverse Fourier transform, which can be approached using techniques from harmonic analysis.
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

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.

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.

fresh_42
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.

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.

## What are central moments?

Central moments are a set of statistical measures that describe the shape, spread, and skewness of a probability distribution.

## What is the significance of central moments?

Central moments are used to quantify the characteristics of a probability distribution, which can help in understanding and analyzing data. They also play a crucial role in statistical modeling and hypothesis testing.

## What is the difference between central moments and raw moments?

The main difference between central moments and raw moments is that central moments are calculated with respect to the mean of a distribution, while raw moments are calculated with respect to the origin (usually 0).

## What does it mean for a central moment to be higher than 4th order?

When a central moment is higher than 4th order, it indicates that the distribution has a high degree of variability or asymmetry. This can be seen as the distribution being more spread out or having a longer tail on one side compared to the other.

## How are central moments used in data analysis?

Central moments are used in data analysis to assess the shape and characteristics of a probability distribution, which can provide insights into the underlying data. They can also be used to compare distributions and identify outliers or anomalies in the data.

• Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
11
Views
10K
• Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
8K
• Differential Equations
Replies
5
Views
3K
• Linear and Abstract Algebra
Replies
16
Views
3K
• Set Theory, Logic, Probability, Statistics
Replies
6
Views
5K
• Astronomy and Astrophysics
Replies
1
Views
1K
• STEM Career Guidance
Replies
4
Views
1K