Normal-inverse Gaussian distribution

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. 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.

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
 * $$x\sim\mathcal{NIG}(\alpha,\beta,\delta,\mu) \text{ and } y=ax+b,$$

then
 * $$y\sim\mathcal{NIG}\bigl(\frac{\alpha}{\left|a\right|},\frac{\beta}{a},\left|a\right|\delta,a\mu+b\bigr).$$

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 the following sense: if $$X_1$$ and $$X_2$$ are independent random variables 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, $$N(\mu,\sigma^2),$$ 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), $$W^{(\gamma)}(t)=W(t)+\gamma t$$, we can define the inverse Gaussian process $$A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.$$ Then given a second independent drifting Brownian motion, $$W^{(\beta)}(t)=\tilde W(t)+\beta t$$, the normal-inverse Gaussian process is the time-changed process $$X_t=W^{(\beta)}(A_t)$$. 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.

As a variance-mean mixture
Let $$\mathcal{IG}$$ denote the inverse Gaussian distribution and $$\mathcal{N}$$ denote the normal distribution. Let $$z\sim\mathcal{IG}(\delta,\gamma)$$, where $$\gamma=\sqrt{\alpha^2-\beta^2}$$; and let $$x\sim\mathcal{N}(\mu+\beta z,z)$$, then $$x$$ follows the NIG distribution, with parameters, $$\alpha,\beta,\delta,\mu$$. This can be used to generate NIG variates by ancestral sampling. It can also be used to derive an EM algorithm for maximum-likelihood estimation of the NIG parameters.