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):
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):
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?).
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.
Background information of the Adjusted Sharpe Ratio (ASR) method: The ASR (Pézier 2004):
The skewness measure by Pézier (2004):
The kurtosis measure by Pézier (2004):
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