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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!

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    $\begingroup$ What might be more natural than the fifth central moment is the fifth cumulant: $\kappa_5 = \mu_5 - 10\mu_3 \mu_2$, where $\mu_k$ is the $k$th central moment. Like the fifth central moment, the fifth cumulant is fifth-degree homogeneous and translation-invariant, but it is also "cumulative", i.e. if $X_1,\ldots,X_n$ are independent random variables then $\kappa_5(X_1+\cdots+X_n) = \kappa_5(X_1) + \cdots + \kappa_5(X_n)$. $\qquad$ $\endgroup$ – Michael Hardy May 6 '16 at 15:39
  • $\begingroup$ Thanks, I was aware of the cumulant function. How does that contribute to an actual interpretation of the hyper skewness? Ideally, I would obtain a similar interpretation as for the skewness/kurtosis. But perhaps that is not possible. $\endgroup$ – Cavents May 6 '16 at 17:32
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    $\begingroup$ Take a look at: en.wikipedia.org/wiki/Moment_%28mathematics%29#Higher_moments $\endgroup$ – Guilherme Thompson May 6 '16 at 18:00
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You can easily and precisely interpret all of those parameters in terms of a histogram or density plot of certain transformations.

  1. First, transform the random variable or the data $X$ to $Z$-scores via $ Z = (X - \mu)/\sigma$.

  2. If you want the $n$-skewness, calculate $V = Z^n$.

  3. Consider the univariate distribution (continuous density, discrete or empirical mass function) of $V$, say $p_V(v)$. Since $n$-skewness $=E(V)$, The point of balance of this distribution is the $n$-skewness.

Note that this representation also provides an easy way to visualize what higher and lower values of $n$-skewness tell you about a distribution (either empirical or theoretical). For example, suppose $X_1$ has 4-skewness (kurtosis) 4.3 and $X_2$ has kurtosis 5.8. What does this tell you about the difference between the distributions of $X_1$ and $X_2$? To answer, graph the distribution of $V_2 = \{(X_2 - \mu_2)/\sigma_2\}^4$, and place a fulcrum at 4.3 on the horizontal axis. The distribution "falls to the right" of the fulcrum, since 5.8 > 4.3. Thus, either or both tails of the distribution of $X_2$ are heavier than those of $X_1$.

Since the curve falls to the right, not to the left, this example shows that kurtosis measures tail weight, and corrects a comment of the original post that kurtosis measures "how peaked the distribution is."

For odd $n$, the hyperskewness measures the relative weight (leverage) of the two tails of the distribution of $X$, with larger $n$ emphasizing points farther out in the tails.

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