Generalised hyperbolic distribution

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 (GIG). 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.

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	${\displaystyle {\frac {e^{\mu z}\gamma ^{\lambda }}{({\sqrt {\alpha ^{2}-(\beta +z)^{2}}})^{\lambda }}}{\frac {K_{\lambda }(\delta {\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}{K_{\lambda }(\delta \gamma )}}}$