Generalised hyperbolic distribution

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generalised hyperbolic

Probability density function
Cumulative distribution function
Parameters	$\mu$ location (real) $\lambda$ (real) $\alpha$ (real) $\beta$ skewness (real) $\delta$ scale (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$
Support	$x \in (-\infty;+\infty)\!$
pdf	$\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} \; e^{\beta (x - \mu)} \!$ $\times \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}} \!$
cdf
Mean	$\mu$
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf	$e^{\mu z} \frac{\gamma^\lambda}{\gamma_z^\lambda} \frac{K_\lambda(\delta \gamma_z)}{K_\lambda (\delta \gamma)}$
Char. func.

The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function

$f(x) = \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}\; \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}}\; e^{\beta (x - \mu)}$

where $K_\nu$ is the modified Bessel function of the second kind.

As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the hyperbolic distribution and the normal-inverse Gaussian distribution.

Its main areas of application are those which require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails, a property that the normal distribution does not possess. The generalised hyperbolic distribution is well-used in economics, with particular application in the fields of modelling financial markets and risk management, due to its semi-heavy tails.

Related distributions

• $X \sim \mathrm{GH}(-\frac{\nu}{2}, 0, 0, \sqrt{\nu}, \mu)$ has a Student's t-distribution with $\nu$ degrees of freedom.
• $X \sim \mathrm{GH}(1, \alpha, \beta, \delta, \mu)$ has a hyperbolic distribution.
• $X \sim \mathrm{GH}(-1/2, \alpha, \beta, \delta, \mu)$ has a normal-inverse Gaussian distribution.