The generalized hyperbolic distribution is said to have semi-heavy tails. I know, that heavy tails means, that the tails are not exponentially bounded, or it has heavier tails than the normal distribution. The german wikipedia connects this to the excess function and knows a medium-tailed distribution. I looked at google books and google, but I could not find a good source, which explains what semi-heavy tail means and especially what is the difference to heavy-tailed and why should it be more appropriate for financial data than the heavy-tailed. I thought heavy-tailed is appropriate for financial data, since we can observe this behaviour empirically.


1 Answer 1


This is explained (curiously in the context of the GH distribution) in the paper


The tail behavior of the GH density is ‘semi-heavy’, i.e. the tails are lighter than those of non-Gaussian stable laws and TSDs [Tempered Stable Distributions] with a relatively small truncation parameter (see Figure 1.4), but much heavier than Gaussian.

Although some financial people are very enthusiastic about using these distributions, it has been shown that Maximum Likelihood Estimation requires thousands of observations to be reliable in this family.


The term financial data is too wide to be discussed in terms of a single family of distributions. However, in this context is common to find data sets containing extreme values which makes necessary to employ distributions with heavier tails that those of the normal ones. Despite this, in vague words, these extreme observations are not too extreme. For this reason people employ this kind of "semi-heavy tailed" distributions.

Modelling is an art, and there is no theorem that tells you the right distribution. The idea is to understand the properties of the distributions in order to employ those whose properties match our intuition or the evidence about a phenomenon of interest.

  • $\begingroup$ In the context of H distribution ( aka heavy tailed LambertW x Gaussian) this is not accurate anymore. MLE achieves unbiased ( empirical) results with as few as 50 observations. The 1000s of observations was necessary due to numerical approximations of inverse of H function. Lambert W x Gaussian provides the closed form inverse, hence no issues for MLE optimization. See github.com/gmgeorg/LambertW and references therein. $\endgroup$ Commented Feb 5, 2022 at 13:47

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