Normal-inverse Gaussian distribution

Normal-inverse Gaussian (NIG)
Parameters	$\mu$ location (real) $\alpha$ tail heaviness (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma =\sqrt{{\alpha }^2-{\beta }^2}$
Support	$x\in (-\mathrm{\infty };+\mathrm{\infty })$
PDF	$\frac{\alpha \delta K_1\left(\alpha \sqrt{{\delta }^2+(x-\mu {)}^2}\right)}{\pi \sqrt{{\delta }^2+(x-\mu {)}^2}}{e}^{\delta \gamma +\beta (x-\mu )}$ $K_j$ denotes a modified Bessel function of the second kind
Mean	$\mu +\delta \beta /\gamma$
Variance	$\delta {\alpha }^2/{\gamma }^3$
Skewness	$3\beta /\sqrt{{\alpha }^2\delta \gamma }$
Excess kurtosis	$3(1+4{\beta }^2/{\alpha }^2)/(\delta \gamma )$
MGF	$e^{\mu z+\delta (\gamma -\sqrt{{\alpha }^2-(\beta +z{)}^2})}$
CF	$e^{i\mu z+\delta (\gamma -\sqrt{{\alpha }^2-(\beta +iz{)}^2})}$

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.

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

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

then

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

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 $\alpha \to \mathrm{\infty }$.

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 $ℐ𝒢$ denote the inverse Gaussian distribution and $𝒩$ denote the normal distribution. Let $z\sim ℐ𝒢(\delta ,\gamma )$, where $\gamma =\sqrt{{\alpha }^2-{\beta }^2}$; and let $x\sim 𝒩(\mu +\beta z,z)$, 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.