Platykurtic and Leptokurtic

[sipanivishal]

Hi David,

Can you provide provide a figure showing when and how they cross each other wrt to Normal distribution.
T distribution is a leptokurtic....does it too have thinner tail than normal ? In your spreadsheet I can see it to cross normal just once making it to have fatter tail than normal.

Thanks
Sipani

 

[David Harper CFA FRM]

Hi Sipani,

Sorry for delay. I don't have hand figure, but see these wikipedia figures:
http://en.wikipedia.org/wiki/Student's_t-distribution

Please see this thread also.

One of the things I have learned this year is that maybe "fat-tailed" is inferior to "heavy-tailed" because leptokurtosis refers to density in the tail and that can still be a long-skinny tail (note leptokurtosis really starts with the idea of peakedness in the middle)

So, the visual shape ultimately is not the critical thing here. What we mean by leptokurtosis, is that at 99% confidence:

the normal is -2.33 standard deviations, but
the t distribution is < -2.33 (but converging to 2.33 with high d.f.)

that's the meaning of leptokurtosis (heave tailed) that, i think, we care about....David

 

[peter333]

Hi David !

Sorry for again asking this but I am not sure whether (i) leptokurtic distributions are high peaked then thinner than normal and finally fatter than normal in tails (ii) platykurtic tails are fatter than normal (iii) platykurtic tails are shorter than leptokurtic but fatter as well.

I actually saw questions comparing these and interestingly I could not google or wiki this topic adequately. Your above- link is not working but wiki diagram does not answers well

best regards,

peter

 

[David Harper CFA FRM]

Hi Peter,

Frankly, I am not conclusive myself on the "semantics" of leptokurtosis. Here is my best understanding (informed by researching Kevin Dowd's book, who is so often precise):

* The common idea (e.g., if you do a google search) is that leptokurtosis refers to peakedness (i.e., higher peak gives fatter tails), and strickly speaking, this appears to be incorrect. As proof, consider the student's t gives confusion (just today this same confusion!) under this visual/semantic definition: it is less peaked than normal, but it is always leptokurtotic (kurtosis > 3). Ergo, higher peak appears to not satisfy.

* I also think "fat/skinny" tail is confusing: a skinny long tail can be leptokurtotic. Fat/skinny gives me confusion b/c it implies vertical girth in the tail.

* Therefore, following Dowd, I am currently best liking: HEAVY TAIL

And, ultimately, the mathematical (density) meaning is the one to trust: the distribution is heavy-tailed if cumulative distribution function (CDF) is less than/greater than the normal. What i mean is, what makes the studen't t heavy (lepto) ultimately is:

Student's CDF (at very low or very high x%) < Normal CDF (at same x%); e.g.,
TINV(1%) < NORMSINV(1%)

From a risk perspective, that's what we care about.

David

 

[peter333]

Thanks David: your explaination is quite helpfull. distributions are somewhat confusing but lets hope the exam will not get too far off-the-point.

best regards,

peter