Central moments

[John Lee]

Hi there

Just posting a question I received from a student as it's really good.

Hello, I am Ioannis , just a quick question on probability theory.

The variance is defined as E[(X - m)^2] and gives a measure of the spread of the returns around the mean.
Why is it wrong to take E [ l X - m l ] or E[(X - m)^4] instead as a measure of the spread?
All three expressions inside the brackets are +ve.

Similarly, the skewness is defined as E[(X - m)^3]. Why is it wrong to take E[(X - m)^5] instead?
What does E[(X - m)^5] represent and why?

I understand how the calculations work. I just want to see the logic behind the formulas.

Many thanks,

Ioannis

 

[bobbathejobba]

The variance is defined as E[(X - m)^2] and gives a measure of the spread of the returns around the mean.
Why is it wrong to take E [ l X - m l ] or E[(X - m)^4] instead as a measure of the spread?
All three expressions inside the brackets are +ve.

It's been a while since I've looked at these, but I would guess that E[|X-m|] is not used as it's messy to integrate - though I tried it out with f(x) = ¾(1-x²) -1<x<1 and got a value of 0.375 compared to a true sd of 0.447.

I think the variance also has more useful statistical properties such as the chi-square distribution whereas a modulus most definitely won't.

Got it! Just tried calculating E(|X-m|) for a U(0,2) and got zero! This will happen for all distributions that are wholly positive or wholly negative as E(|X-m|) = E(X-m) = E(X)-m = m - m = 0

The 4th central moment is the kurtosis of a distribution and doesn't give the spread but the curvature of the distribution. See here for details.

Similarly, the skewness is defined as E[(X - m)^3]. Why is it wrong to take E[(X - m)^5] instead?
What does E[(X - m)^5] represent and why?

I have no idea what the fifth moment gives - but looking at the progression it seems that it will give a different feature to the asymmetry. It still gives 0 for a symmetrical distribution but will be more weighted by extreme values.

General details about moments can be found here.

 

[Actuary]

There's a paragraph in the notes (Ch1 pg13) which refers to absolute function as "not a nice function to deal with mathematically - in particular ... has a nasty kink when x=0, whereas graph of x^2 is always a nice "smooth" curve."

A bit fluffy but gets the point across. It represents spread in the simplest way.

Higher order moments describe different things as bob says. E[(X - m)^5] represents the fifth central moment.