Generalised hyperbolic distribution

Generalised hyperbolic

Parameters	$\lambda$ (real) $\alpha$ (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\mu$ location (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$
Support	$x \in (-\infty; +\infty)\!$
PDF	$\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} \; e^{\beta (x - \mu)} \!$ $\times \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}} \!$
Mean	$\mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}$
Variance	$\frac{\delta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left( \frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)} - \frac{K_{\lambda+1}^2(\delta\gamma)}{K_{\lambda}^2(\delta\gamma)} \right)$
MGF	$\frac{e^{\mu z}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +z)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +z)^2})}{K_\lambda (\delta \gamma)}$

The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution. Its probability density function (see the box) is given in terms of modified Bessel function of the second kind, denoted by $K_\lambda$. It was introduced by Ole Barndorff-Nielsen, who studied it in the context of physics of wind-blown sand.

Properties

Linear transformation

This class is closed under affine transformations.

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution has Infinite divisibility and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the GIG distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinite divisible as well.

Related distributions

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

• $X \sim \mathrm{GH}(-\frac{\nu}{2}, 0, 0, \sqrt{\nu}, \mu)\,$ 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 (NIG).
• $X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,$ normal-inverse chi-squared distribution
• $X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,$ normal-inverse gamma distribution (NI)

• $X \sim \mathrm{GH}(\lambda, \alpha, \beta, 0, \mu)\,$ has a variance-gamma distribution.

Applications

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