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

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 (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

Hyperbolic distribution

The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal 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.

Related distributions

• has a Student's t-distribution with degrees of freedom.
• has a hyperbolic distribution
  Hyperbolic distribution
  The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution...
  
  .

• has a normal-inverse Gaussian distribution (NIG).
• normal-inverse chi-squared distribution
• normal-inverse gamma distribution (NI)

• has a variance-gamma distribution.