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# Error in Miller's illustration of leptokurtosis

#### brian.field

##### Well-Known Member
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A distribution with positive (excess) kurtosis (leptokurtosis) is MORE peaked than a normal distribution with lower probabilities at (around) plus and minus 1 standard deviation AND more probabilities in the tails, i.e., fatter tails than the normal distribution. The picture presenting kurtosis in Miller's text (on page 51) exhibits fat tails but erroneously does NOT exhibit the peaked ness that is pertinent.

Just a heads up to everyone.....and obviously, if I am mistaken, please let me know.

(Actually, the illustration of low kurtosis on page 52 is similarly flawed, I believe.)

Best,

Brian

#### IRENA BLAZINCIC

##### Member
Are you referring to picture that shows Student's t probability density functions with different degrees of freedom (k)? (my book is 2013 edition)
I think that in order to see the peakedness you should compare two distributions with the same variance. Variance of the student's t distribution for k>2 is k/(k-2).
If you compare student t-distribution to standard normal distribution (variance =1), student t-distribution should be scaled to varinace 1. And then the student t distribution would be more peaked than standard normal distribution.

#### brian.field

##### Well-Known Member
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The pages I quoted were associated with the second edition but I believe the pictures are the same in both editions, but the page numbers may differ.

Also, what you are saying seems reasonable, irenab (thank you).

Still, I think the illustration was meant to compare a normal distribution, which is the dotted line and reflects "no excess kurtosis," and a distribution with high kurtosis, which I interpret to mean positive excess kurtosis. The solid line is labelled as high kurtosis but does not reflect this fact simply because the curve is LESS peaked than the no excess kurtosis curve.

Yet, you bring up an interesting question. I know that the t-distribution is leptokurtotic, which means that the distribution has excess kurtosis. Also, the t-distribution approaches the normal distribution as n approaches infinity, so I deduce that as n approaches infinity, the excess kurtosis for the t-distribution approaches 0. I also deduce from this that the t-distribution will be leptokurtotic for all n less than infinity, so it should alwaysbe more peaked and fatter tails than the normal distribution. Now here is the rub; I have always been told that the t-distribution is similar to the normal distribution but is flattened somewhat, as if pressure was applied to the peak and probability was spread to the tails. (Most pictures of the t indicate the same.)

But if this is the case, then it is contradictory to say that the t-distribution is leptokurtotic.....essentially, it comes down to this: My understanding of leptokurtosis is that the peak of the distribution is HIGHER and SKINNIER and the tails are FATTER versus the normal distribution. Based on the t-distributions I have seen, it appears impossible to have a Higher peak, and therefore, impossible to have excess kurtosis.....

Anyone care to opine

Thanks!

Brian

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#### brian.field

##### Well-Known Member
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Ok, the image below presents my understanding of Leptokurtosis....(and the corresponding inconsistency with the t-distribution.)

David - any thoughts?

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#### David Harper CFA FRM

##### David Harper CFA FRM
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HI @brianhfield

Love you diagram!

I agree that Miller's plot choice is unfortunate, but authors occasionally do select this rendering (i.e., where the student's t appears less peaked). As @irenab writes, implicitly he is rescaling neither the normal nor the student's t (although Miller's plots are stylistic in the first place: there are not y pdf values). Without any adjustment, the student's t will appear less peaked, but at the same time, it will have a variance = df/(df-2) > 1. If normal the normal is rescaled to match the variance (an apples-to-apples for the second moment, if you will), the expected higher peakedness of the student's t will be revealed; i.e., if the variances match, the comparison will look like your diagram!

Okay, but it turns out that kurtosis does not (100%) correspond to both higher peaks and heavier tails necessarily; rather, it is just the majority use case and the intuitive expectation. See http://stats.stackexchange.com/questions/80626/kurtosis-of-made-up-distribution

Here is a recommended paper, "On the Meaning and Use of Kurtosis"
https://www.dropbox.com/s/2vwgo9e826k4z5g/DeCarlo_OnMeaningUseKurtosis.pdf?dl=0
For example,
Why are tailedness and peakedness both components of kurtosis? It is basically because kurtosis represents a movement of mass that does not affect the variance. Consider the case of positive kurtosis, where heavier tails are often accompanied by a higher peak. Note that if mass is simply moved from the shoulders of a distribution to its tails, then the variance will also be larger. To leave the variance unchanged, one must also move mass from the shoulders to the center, which gives a compensating decrease in the variance and a peak. For negative kurtosis, the variance will be unchanged if mass is moved from the tails and center of the distribution to its shoulders, thus resulting in light tails and flatness."

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#### agattik

##### New Member
This is also well discussed here https://en.wikipedia.org/wiki/Kurtosis#Interpretation. The exercise in question (203.3) appears misleading to me since the analyst's conclusion makes sense only if we assume both portfolios have same variance (well explained in the paper "On the Meaning and Use of Kurtosis"). Since there is no data indicating same variance, there appears to be no support for the analyst's conclusion about peakedness. That said, neither can we conclude based on the data that "the analyst is wrong" It might be good to change the question to say the portfolios have the same variance.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @agattik Thank you, the source question (P1.T2.203) is here at https://www.bionicturtle.com/forum/threads/p1-t2-203-skew-and-kurtosis-stock-watson.5223/
... But is the qualifier "same variance" a sufficient condition? Yes, that would imply peakedness for a student's distribution, but the question doesn't state an assumption for the distribution. It's not clear to me that standardizing variance guarantees peakedness when the distribution has kurtosis > 3. My read of DeCarlo is that there exist exceptions ... a side issue here is that I will get ahead of GARP: the question is based on a GARP question and the naive understanding w.r.t. peakedness. Thanks,