Dear Adam,

in my opinion, the skewness and kurtosis parameters may be sufficient or not, depending if you know the PDF explicitly.

For example, a Rayleigh distribution is fully described by 2 parameters and it is non-Gaussian with sk=0.631, ku=0.245 (fixed). In this case, sk and ku fully describe the deviation from Gaussian PDF. If you take a three-parameter PDF (Weibull?), the sk and ku would be also sufficient to describe deviation from Gaussianity (you have 3 dof in your PDF).

By following this approach, I can imagine to build a PDF with 4 parameters and to calibrate them to obtain given values of sk and ku, which then fully describe the PDF (i.e. the solution is unique). With 5 parameters or more, instead, the solution would not be unique anymore.

If you do not know the PDF equation explicitly, sk and ku are only approximations, as I think that you can have infinite PDFs with the same values of sk and ku. In fact, such parameters derive from the PDF moments, which are integral properties of the PDF (infinite functions may be solution of the same integral). For example, one interesting problem in probability is to reconstruct a PDF from a given set of PDF moments.

In this latter case, sk adn ku are not sufficient and you need the whole PDF to get a full description of non-Gaussian behaviour. However, as the knowledge of the whole PDF could be unfeasible from practical reasons, one often relies to sk and ku only.

This is my guess. I hope it helps.

Denis

Thank you, Denis, for your answer. As always, I can count on you. I really like your detailed answer, but I will continue to look for additional parameters describing the distribution without the knowledge of the shape of it. This will be great for us to have something like second order skewness or kurtosis :-)

Maybe some proposition?

Adam