Normal-inverse Gaussian distribution

Normal-inverse Gaussian (NIG)
Parameters	$\mu$ location (real) $\alpha$ tail heavyness (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}$
Support	$x\in (-\infty ;+\infty )\!$
PDF	${\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}$ $K_{j}$ denotes a modified Bessel function of the second 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	$e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}$
CF	$e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}$

The normal-inverse Gaussian distribution (NIG) is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian 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 normal-inverse Gaussian 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 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 $\delta _{1}+\delta _{2}.$

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 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

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