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generalised hyperbolic
Probability density function
Cumulative distribution function
Parameters location (real)
(real)
(real)
skewness (real)
scale (real)
Support
pdf
cdf
Mean
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf
Char. func.

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

where 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 distributions

  • has a Student's t-distribution with degrees of freedom.
  • has a hyperbolic distribution.
  • has a normal-inverse Gaussian distribution.


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