Normal-inverse Gaussian distribution

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 third kind
Mean	${\displaystyle \mu +\delta \beta /\gamma }$
Variance	${\displaystyle \delta \alpha ^{2}/\gamma ^{3}}$
Skewness	${\displaystyle 3\beta /(\alpha {\sqrt {\delta \gamma }})}$
Ex. 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 normal-inverse Gaussian distribution (NIG) 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.

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

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

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: 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 ${\displaystyle \delta _{1}+\delta _{2}.}$

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