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Generalised hyperbolic distribution

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generalised hyperbolic
Probability density function
Cumulative distribution function
Parameters \mu location (real)
\lambda (real)
\alpha (real)
\beta skewness (real)
\delta scale (real)
\gamma = \sqrt{\alpha^2 - \beta^2}
Support x \in (-\infty; +\infty)\!
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}} \!
Mean \mu
mgf e^{\mu z} \frac{\gamma^\lambda}{\gamma_z^\lambda} \frac{K_\lambda(\delta \gamma_z)}{K_\lambda (\delta \gamma)}
Char. func.

The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function

f(x) = \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}\;
\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}}\; e^{\beta (x - \mu)}

where K_\nu is the modified Bessel function of the second kind.

As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the hyperbolic distribution and the normal-inverse Gaussian 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.

Related distributionsEdit

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