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The normal-inverse Gaussian distribution (NIG) 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 ...
that is defined as the normal variance-mean mixture where the mixing density is the
inverse Gaussian distribution In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞). Its probability density function is given by : f(x;\mu, ...
. The NIG distribution was noted by Blaesild in 1977 as a subclass of the
generalised hyperbolic distribution 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 (GIG). Its probability density functi ...
discovered 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 ...
. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that requir ...
literature in 1997. The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.


Properties


Moments

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.


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, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If : x\sim\mathcal(\alpha,\beta,\delta,\mu) \text y=ax+b, then : y\sim\mathcal\bigl(\frac,\frac,\left, a\\delta,a\mu+b\bigr).


Summation

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.


Convolution

The class of normal-inverse Gaussian distributions is closed under
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
in the following sense: if X_1 and X_2 are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s that are NIG-distributed with the same values of the parameters \alpha and \beta, but possibly different values of the location and scale parameters, \mu_1, \delta_1 and \mu_2, \delta_2, respectively, then X_1 + X_2 is NIG-distributed with parameters \alpha, \beta, \mu_1+\mu_2 and \delta_1 + \delta_2.


Related distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and 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 ...
, N(\mu,\sigma^2), arises as a special case by setting \beta=0, \delta=\sigma^2\alpha, and letting \alpha\rightarrow\infty.


Stochastic process

The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It i ...
), W^(t)=W(t)+\gamma t, we can define the inverse Gaussian process A_t=\inf\. Then given a second independent drifting Brownian motion, W^(t)=\tilde W(t)+\beta t, the normal-inverse Gaussian process is the time-changed process X_t=W^(A_t). The process X(t) at time t=1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of
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 ...
.


As a variance-mean mixture

Let \mathcal denote the
inverse Gaussian distribution In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞). Its probability density function is given by : f(x;\mu, ...
and \mathcal denote 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 ...
. Let z\sim\mathcal(\delta,\gamma), where \gamma=\sqrt; and let x\sim\mathcal(\mu+\beta z,z), then x follows the NIG distribution, with parameters, \alpha,\beta,\delta,\mu. This can be used to generate NIG variates by
ancestral sampling An ancestor, also known as a forefather, fore-elder or a forebear, is a parent or ( recursively) the parent of an antecedent (i.e., a grandparent, great-grandparent, great-great-grandparent and so forth). ''Ancestor'' is "any person from who ...
. It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.


References

{{DEFAULTSORT:Normal-Inverse Gaussian Distribution Continuous distributions