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

${\displaystyle K_{\lambda }}$

Properties

Linear transformation

This class is closed under affine transformations.^[1]

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible 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 infinitely divisible as well.^[3]

Fails to be convolution-closed

An important point about infinitely divisible distributions is their connection to

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.

• ${\displaystyle \mathrm {GH} (-{\frac {\nu }{2}},0,0,{\sqrt {\nu }},\mu )\,}$ is a Student's t-distribution with ${\displaystyle \nu }$ degrees of freedom.
• ${\displaystyle \mathrm {GH} (1,\alpha ,\beta ,\delta ,\mu )\,}$ is a hyperbolic distribution.

• ${\displaystyle \mathrm {GH} (-1/2,\alpha ,\beta ,\delta ,\mu )\,}$ is a normal-inverse Gaussian distribution (NIG).
• ${\displaystyle \mathrm {GH} ({\text{?}},{\text{?}},{\text{?}},{\text{?}},{\text{?}})\,}$ normal-inverse chi-squared distribution
• ${\displaystyle \mathrm {GH} ({\text{?}},{\text{?}},{\text{?}},{\text{?}},{\text{?}})\,}$
  normal-inverse gamma distribution
  (NI)
• ${\displaystyle \mathrm {GH} (\lambda ,\alpha ,\beta ,0,\mu )\,}$ is a variance-gamma distribution
• ${\displaystyle \mathrm {GH} (1,1,0,0,\mu )\,}$ is a Laplace distribution with location parameter ${\displaystyle \mu }$ and scale parameter 1.

Applications

It is mainly applied to areas that require sufficient probability of far-field behaviour^[