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I want to calculate the skewness and the excess kurtosis (third and fourth moment) of a (trading rule) return distribution. To calculate the skewness coefficient I'm using the quantile-based skewness measure of Hinkley (1975):

Skewness measure

Now I'm trying to find a robust quantile-based kurtosis measure (for asymmetric distributions, like the trading rule return distribution). I found one introduced by Ruppert (1987):

Kurtosis measure

However I'm not sure how to apply this (/write this as the same way of the skewness measure) on my return distribution? (What does the greek letter "eta" precisely mean in this situation?).

References:

Hinkley, D. V. (1975). On power transformations to symmetry. Biometrika, 62(1), 101-111.

Ruppert, D. (1987). What is kurtosis? An influence function approach. The American Statistician, 41(1), 1-5.

EDIT:

Background information of the Adjusted Sharpe Ratio (ASR) method: The ASR (Pézier 2004): ASR

The skewness measure by Pézier (2004): Skewness

The kurtosis measure by Pézier (2004): Kurtosis

Reference: Pezier, J., Alexander, C., & Sheedy, E. (2004). Risk and risk aversion. Alexander C. & Sheedy E. The Professional Risk Managers’ Handbook, 1. Direct link: Handbook

Page 65: Adjusted Sharpe Ratio

Pages 708-711: Skewness and Kurtosis measure

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The denominator is essentially just some kind of "middle" part of the distribution for the tail to be large or not so large relative to; you need something to scale the $p$-distance by. For my illustrations here I chose the middle half (interquartile range) as a reasonable default; it's the most obvious one to try.

So let's take say $p=0.01$ and $\eta=0.25$ (I don't claim that's a n ideal choice, but $\eta=0.25$ seems a reasonable default).

Then $k = R_{0.25,0.01}= \frac{q_{0.99}-q_{0.01}}{q_{0.75}-q_{0.25}}$.

This then will be larger when the tail is heavy (since the small $p$ with a heavy tail will make the numerator large) and large when the distribution is peaked (since the larger $\eta$ with a peaked distribution will make the denominator small).

Here's an illustration for the normal:

enter image description here

For that particular pair $\eta,p$ we have:

 distribution     k
   uniform       1.96
   normal        3.45
   Cauchy        31.8
   exponential   4.18

You can of course choose different values for $\eta$ and $p$.

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  • $\begingroup$ Indeed (I take $p=0.01$ as well). However the struggle is in the denominator part of the formula for me. How do you define/interpret $\eta$? And what number do you assign to it (how to determine it)? I couldn't find a clear explanation. $\endgroup$ – Wildman Oct 28 '15 at 9:03
  • $\begingroup$ @Sander See my edit that posted right after your comment. It is essentially just some kind of "middle" part of the distribution for the tail-range to be large or small relative to; you need something to scale the p-distance by. I chose the middle half as a reasonable default; it's the most obvious one to try. $\endgroup$ – Glen_b -Reinstate Monica Oct 28 '15 at 9:05
  • $\begingroup$ I understand now, thanks! Is there some "official" definition/literature for $\eta$ in this case (or is it just some scaling number)? Indeed the middle half sounds like a reasonable default. $\endgroup$ – Wildman Oct 28 '15 at 9:17
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    $\begingroup$ To echo the suggestion of @kjetil b halvorsen (see comment on question), an $L$-moment measure of kurtosis is a weighted combination of all quantiles and avoids all arbitrariness of which inner and which outer pair you use. It's not going to be robust/resistant to really fat or long tails, but $L$-moments people regard this as fair enough: it's going to measure what it is supposed to. (It be much, much more robust/resistant than moment-based kurtosis.) $\endgroup$ – Nick Cox Oct 28 '15 at 12:27
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    $\begingroup$ Nassim Taleb's free book Silent Risk (available from his website) has some deep meditations on risk as a function of extreme values in returns that includes developing a number of quite practical metrics for tail risk. $\endgroup$ – Mike Hunter Oct 28 '15 at 13:59

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