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	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$
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 modifiedBessel function of the second kind[1]
Mean	${\displaystyle \mu +\delta \beta /\gamma }$
Variance	${\displaystyle \delta {\alpha }^2/{\gamma }^3}$
Skewness	${\displaystyle 3\beta /\sqrt{{\alpha }^2\delta \gamma }}$
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	${\displaystyle e^{i\mu z+\delta (\gamma -\sqrt{{\alpha }^2-(\beta +iz{)}^2})}}$

Thenormal-inverse Gaussian distribution (NIG, also known as thenormal-Wald distribution) is acontinuous probability distribution that is defined as thenormal variance-mean mixture where the mixing density is theinverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of thegeneralised hyperbolic distribution discovered byOle Barndorff-Nielsen.^[2] In the next year Barndorff-Nielsen published the NIG in another paper.^[3] It was introduced in themathematical finance literature in 1997.^[4]

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

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

This class is closed underaffine transformations, since it is a particular case of theGeneralized hyperbolic distribution, which has the same property. If

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

then^[8]

${\displaystyle y\sim \mathcal{N}\mathcal{I}\mathcal{G}(\frac{\alpha }{\left|a\right|},\frac{\beta }{a},\left|a\right|\delta ,a\mu +b).}$

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

The class of normal-inverse Gaussian distributions is closed underconvolution in the following sense:^[9] if${\displaystyle X_1}$ and${\displaystyle X_2}$ areindependentrandom 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 thenormal distribution,${\displaystyle N(\mu ,{\sigma }^2),}$ arises as a special case by setting${\displaystyle \beta =0,\delta ={\sigma }^2\alpha ,}$ and letting${\displaystyle \alpha \to \mathrm{\infty }}$.

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 ofLévy processes.

Let${\displaystyle \mathcal{I}\mathcal{G}}$ denote theinverse Gaussian distribution and${\displaystyle \mathcal{N}}$ denote thenormal distribution. Let${\displaystyle z\sim \mathcal{I}\mathcal{G}(\delta ,\gamma )}$, 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 byancestral sampling. It can also be used to derive anEM algorithm formaximum-likelihood estimation of the NIG parameters.^[10]

• Continuous distributions

• Articles with short description
• Short description is different from Wikidata