Normal-inverse Gaussian distribution

normal-inverse Gaussian (NIG)

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

The normal-inverse Gaussian distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. This distribution is often referred to as the NIG-distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distributions.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If $X_{1}$ and $X_{2}$ are independent random variable 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 $\delta _{1}+\delta _{2}.$