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.^[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]

Normal-inverse Gaussian (NIG)

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

The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.^[5]

Properties

Moments

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.^[6][7]

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

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

then^[8]

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

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:^[9] if ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are independent random variables that are NIG-distributed with the same values of the parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, but possibly different values of the location and scale parameters, ${\displaystyle \mu _{1}}$, ${\displaystyle \delta _{1}}$ and ${\displaystyle \mu _{2},}$ ${\displaystyle \delta _{2}}$, respectively, then ${\displaystyle X_{1}+X_{2}}$ is NIG-distributed with parameters ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$${\displaystyle \mu _{1}+\mu _{2}}$ and ${\displaystyle \delta _{1}+\delta _{2}.}$

Related distributions

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

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), ${\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t}$, we can define the inverse Gaussian process ${\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.}$ Then given a second independent drifting Brownian motion, ${\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t}$, the normal-inverse Gaussian process is the time-changed process ${\displaystyle X_{t}=W^{(\beta )}(A_{t})}$. The process ${\displaystyle X(t)}$ at time ${\displaystyle t=1}$ 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 ${\displaystyle {\mathcal {IG}}}$ denote the inverse Gaussian distribution and ${\displaystyle {\mathcal {N}}}$ denote the normal distribution. Let ${\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )}$, where ${\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$; and let ${\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)}$, then ${\displaystyle x}$ follows the NIG distribution, with parameters, ${\displaystyle \alpha ,\beta ,\delta ,\mu }$. 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.^[10]