Normal-inverse Gaussian distribution


The normal-inverse Gaussian 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. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance 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 transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If
then

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 the following sense: if and are independent random variables that are NIG-distributed with the same values of the parameters and, but possibly different values of the location and scale parameters, , and , respectively, then is NIG-distributed with parameters and

Related distributions

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 and letting.

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,, 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. The process at time has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

As a variance-mean mixture

Let denote the inverse Gaussian distribution and denote the normal distribution. Let, where ; and let, then follows the NIG distribution, with parameters,. 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.