Generalised hyperbolic distribution

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generalised hyperbolic
Probability density function
Cumulative distribution function
Parameters μ location (real)
λ (real)
α (real)
β asymmetry parameter (real)
δ scale parameter (real)
\gamma = \sqrt{\alpha^2 - \beta^2}
Support x \in (-\infty; +\infty)\!
Probability density function (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}} \!
Cumulative distribution function (cdf)
Mean \mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}
Median
Mode
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)
Skewness
Excess kurtosis
Entropy
Moment-generating function (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)}
Characteristic function

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 third kind, denoted by Kλ.

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.

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. This class is closed under linear operations. It was introduced by Ole Barndorff-Nielsen.

[edit] Related distributions

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Probability distributions
 
Continuous univariate supported on the whole real line