Generalised hyperbolic distribution

Generalised hyperbolic

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

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 ${\displaystyle K_{\lambda }}$. It was introduced by Ole Barndorff-Nielsen, who studied it in the context of physics of wind-blown sand.

This class is closed under affine transformations.

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 generalized inverse Gaussian distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinite divisible as well.

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.

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