The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) 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 in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen.

[2] In the next year Barndorff-Nielsen published the NIG in another paper.

[3] It was introduced in the mathematical finance literature in 1997.

[4] The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.

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

[6][7] This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

If then[8] This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:[9] if

are independent random variables that are NIG-distributed with the same values of the parameters

, but possibly different values of the location and scale parameters,

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution,

arises as a special case by setting

β = 0 , δ =

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),

, we can define the inverse Gaussian process

Then given a second independent drifting Brownian motion,

, the normal-inverse Gaussian process is the time-changed process

has the normal-inverse Gaussian distribution described above.

The NIG process is a particular instance of the more general class of Lévy processes.

denote the inverse Gaussian distribution and

denote the normal distribution.

( μ + β z , z )

follows the NIG distribution, with parameters,

α , β , δ , μ

This can be used to generate NIG variates by ancestral sampling.

It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.