Let $X$ be a random variable. The standardized $n$th moment of $X$ is defined as $$\frac{E[(X-\mathbb{E}[X])^n]}{\mbox{Var}[X]^{n/2}}. $$ Special cases are the skewness ($k=3$) and the kurtosis $k=4$. The skewness is a measure for the asymmetry of a distribution while the kurtosis measures how peaked the distribution is. In my financial engineering project, I have to work with $k=5$ which is referred to as hyper skewness. As a benchmark, it is common to consider the normal distribution which has zero skewness and kurtosis equal to three. However, the hyper skewness of the normal distribution is also equal to zero, so at first sight it does not tell anything more about the distribution.
I was wondering if there is a useful interpretation of the hyper skewness? Does someone know any literature about this feature? If there is no any available literature then I can perhaps compute the hyper skewness for a variety of distributions and try to find an interpretation. However, some known literature would spare me some time.
Thanks in advance!