Fat Tails vs Skinny Tails vs Everything in b/w

[Jiew Kwang]

Hi guys,

I have some burning questions about the subject mentioned above.

Firstly, is a normal distribution with kurtosis of 3 considered a skinny or normal tailed distribution?

How can i also visualize leptokurtosis as being fat tailed? As df increases or decreases beyond the normal distribution, how does this translate to distribution becoming fat or skinny tailed?

Does skinny tailed = light tailed and heavy tailed = fat tailed?

Wonder why there are so many definitions. I know it is easier than it sounds but I want to clarify.. Thanks!

regards,
jk

 

[David Harper CFA FRM]

Hi Jiew,

We've had some discussion on this, perhaps inconclusive; e.g., http://forum.bionicturtle.com/viewthread/1554/

Re: is a normal distribution with kurtosis of 3 considered a skinny or normal tailed distribution?
Yes, normal has kurtosis = 3 and this is consider NEITHER light/skinny/platykurtic NOR heavy/fat/leptokurtic but "normal"
… I guess normal is mesokurtic (http://en.wikipedia.org/wiki/Kurtosis) but I never see that term actually used

What you always read is: leptokurtosis signifies a HIGH PEAK which implies a HEAVY TAIL. This frankly does not work personally for me (I feel I can visualize a high peak without heavy tail density!)
…. Although, please note, the student's t is typically displayed, in a sort of optical illusion with a shorter-than-normal peak. However, if you standardize the variance, the student's t will always be peaked (versus normal) as it is always leptokurtic (i.e., kurtosis > 3 or excess kurtosis > 0).

Re: Is skinny tailed = light tailed and heavy tailed = fat tailed
Yes. Although, FWIW, I have come to favor HEAVY-tail because ultimately this is about TAIL DENSITY not (e.g.) the height of the PDF.

My favorite way to define heavy tail is per the math. Consider the normal versus student's t.
First the normal, what is the 99% normal deviate? 2.33
This means, for a normal (normal tail) distribution, the 1% outcome is 2.33 deviates away from mean.
A heavy tail distribution, then, at the 1% tail quantile, must (simply) have a deviate that exceeds 2.33. That's how I think about it.

Student's t at 1% (e.g., df = 18) = 2.567
So, this student's t has a 1% outcome that is 2.567 deviates from mean: the 1% is further into the extreme. It has a heavy tail.

Or, similarly, we can solve for the 1% student's t at the same 2.33 deviate: =TDIST(2.33, 18 df,1) = 1.58%
… whereas a normal 2.33 corresponds to tail with 1% "weight," the student's t has a tail that "weighs" 1.58% at 2.33 deviates (more probability to exceed).

So, we have two ways to express heavy-tail:
* For the same deviate (2.33), the heavy-tail must have more tail probability (1.58% versus 1.0%), or
* For the same tail probability (quantile; e.g., 1.0%), the heavy-tail must have a greater deviate (2.6 versus 2.33)

In this way, my own view is that HEAVY is best, but given that visualization of continuous PDF can be very un-intuitive (it is easy to forget that 1. that height of the pdf function has little visual meaning itself and 2. scales vary), I try not to visualize and focus on mathy significance above.

David

 

[Jiew Kwang]

david,

thanks for that very clear explaination. really solved my doubts.

just a minor thing i'd thought i'd say regarding the quant notes 2010 pg125.

the graphical stegosaurus is a normal distribution but has been characterized "skinny tails". (which partly fueled my confusion earlier)

Thanks!

 

[David Harper CFA FRM]

Jiew - oh, i see your point, thank you ... we may need to retire the stegosaurus. thanks, David