The generalised hyperbolic distribution (GH) is a
continuous probability distribution defined as the
normal variance-mean mixture In probability theory and statistics, a normal variance-mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form
:Y=\alpha + \beta V+\sigma \sqrtX,
where \alpha, \beta and \sigma ...
where the mixing distribution is the
generalized inverse Gaussian distribution
In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
:f(x) = \frac x^ e^,\qquad x>0,
where ''Kp'' is a mod ...
(GIG). Its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
(see the box) is given in terms of
modified Bessel function of the second kind, denoted by
.
[Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013] It was introduced by
Ole Barndorff-Nielsen, who studied it in the context of physics of
wind-blown sand.
Properties
Linear transformation
This class is closed under
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
s.
Summation
Barndorff-Nielsen and Halgreen proved that the GIG distribution is
infinitely divisible and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the
generalized inverse Gaussian distribution
In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
:f(x) = \frac x^ e^,\qquad x>0,
where ''Kp'' is a mod ...
, Barndorff-Nielsen and Halgreen showed the GH distribution is infinitely divisible as well.
Fails to be convolution-closed
An important point about infinitely divisible distributions is their connection to
Lévy processes, i.e. at any point in time a Lévy process is infinitely divisible distributed. Many families of well-known infinitely divisible distributions are so-called convolution-closed, i.e. if the distribution of a Lévy process at one point in time belongs to one of these families, then the distribution of the Lévy process at all points in time belong to the same family of distributions. For example, a Poisson process will be Poisson distributed at all points in time, or a Brownian motion will be normally distributed at all points in time. However, a Lévy process that is generalised hyperbolic at one point in time might fail to be generalized hyperbolic at another point in time. In fact, the generalized Laplace distributions and the normal inverse Gaussian distributions are the only subclasses of the generalized hyperbolic distributions that are closed under convolution.
Related distributions
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
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
, the
hyperbolic distribution, the
normal-inverse Gaussian distribution and the
variance-gamma distribution
The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The ...
.
*
has a
Student's ''t''-distribution with
degrees of freedom.
*
has a
hyperbolic distribution.
*
has a
normal-inverse Gaussian distribution (NIG).
*
normal-inverse chi-squared distribution
*
normal-inverse gamma distribution (NI)
*
has a
variance-gamma distribution
The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The ...
*
has a
Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
with location parameter
and scale parameter 1.
Applications
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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
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.
References
{{DEFAULTSORT:Generalised Hyperbolic Distribution
Continuous distributions