Hi @Mdeclercq Apologies for the confusing statement: I think that's a legacy statement from an FRM syllabus source that

Here is a good thread https://www.bionicturtle.com/forum/threads/error-in-millers-illustration-of-leptokurtosis.7685/ e.g.,

For several reasons including technical accuracy, my preferred short description is simply that leptokurtosis signifies "heavy tails" (a runner-up would be "fat tails," but this can also lead to subtle misunderstandings vis a vis thin versus fat. We will fix this on the next revision. Thank you!

btw, here is a recent video of mine on kurtosis where I say that while kurtosis

https://www.bionicturtle.com/forum/threads/t2-7-kurtosis-of-a-probability-distribution.21738/

**does need to be corrected**(I have made a note, hence your placement here in Errors is appropriate). I haven't seen this paper, but I completely agree that the most accurate definition of kurtosis refers to tail density (as I have written in this forum many times; e.g., https://www.bionicturtle.com/forum/threads/kurtosis-and-peakedness.4758/post-12546).Here is a good thread https://www.bionicturtle.com/forum/threads/error-in-millers-illustration-of-leptokurtosis.7685/ e.g.,

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/DeCarl:confused:nMeaningUseKurtosis.pdf?dl=0

For example,

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/DeCarl:confused:nMeaningUseKurtosis.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."

For several reasons including technical accuracy, my preferred short description is simply that leptokurtosis signifies "heavy tails" (a runner-up would be "fat tails," but this can also lead to subtle misunderstandings vis a vis thin versus fat. We will fix this on the next revision. Thank you!

btw, here is a recent video of mine on kurtosis where I say that while kurtosis

*tends*to associate with peakedness (i explain why this is the case and, in my experience, it is the tendency) the better definition of leptokurtosis is**heavy tails**https://www.bionicturtle.com/forum/threads/t2-7-kurtosis-of-a-probability-distribution.21738/

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