Generalised hyperbolic distribution
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asymmetry parameter (real)
scale parameter (real)
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 .
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.
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. This class is closed under linear operations. It was introduced by Ole Barndorff-Nielsen.
- has a normal-inverse Gaussian distribution (NIG).
- normal-inverse chi-squared distribution
- normal-inverse gamma distribution (NI)
- has a variance-gamma distribution.