The generalised hyperbolic distribution (GH) is a

continuous probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

defined as the 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 mo ...

(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) c ...

(see the box) is given in terms of

modified Bessel function of the second kind Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

, denoted by

K_\lambda

.Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 It was introduced by

Ole Barndorff-Nielsen Ole Eiler Barndorff-Nielsen (18 March, 1935 – 26 June, 2022) was a Denmark, Danish statistician who has contributed to many areas of statistics, statistical science. Education and career He was born in Copenhagen, and became interested in st ...

, 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 ...

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution is

infinitely divisible Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter ...

and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the

, 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 Levy, Lévy or Levies may refer to: People * Levy (surname), people with the surname Levy or Lévy * Levy Adcock (born 1988), American football player * Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker * Levy Fi ...

, 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 expo ...

, the

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

, the

normal-inverse Gaussian distribution The normal-inverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild ...

and the variance-gamma distribution. *

X \sim \mathrm(-\frac, 0, 0, \sqrt, \mu)\,

has a Student's ''t''-distribution with

\nu

degrees of freedom. *

X \sim \mathrm(1, \alpha, \beta, \delta, \mu)\,

has a

. *

X \sim \mathrm(-1/2, \alpha, \beta, \delta, \mu)\,

has a

(NIG). *

X \sim \mathrm(?, ?, ?, ?, ?)\,

normal-inverse chi-squared distribution *

X \sim \mathrm(?, ?, ?, ?, ?)\,

normal-inverse gamma distribution In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distributio ...

(NI) *

X \sim \mathrm(\lambda, \alpha, \beta, 0, \mu)\,

has a variance-gamma distribution *

X \sim \mathrm(1, 1, 0, 0, \mu)\,

has a

with location parameter

\mu

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 i ...

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