Moments edit

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 edit

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 edit

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

Convolution edit

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

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 ${\displaystyle \alpha \rightarrow \infty }$ .

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.

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]