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# Normal-inverse Gaussian distribution

Parameters ${\displaystyle \mu }$ location (real) ${\displaystyle \alpha }$ tail heaviness (real) ${\displaystyle \beta }$ asymmetry parameter (real) ${\displaystyle \delta }$ scale parameter (real) ${\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$ ${\displaystyle x\in (-\infty ;+\infty )\!}$ ${\displaystyle {\frac {\alpha \delta K_{1}\left(\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}\right)}{\pi {\sqrt {\delta ^{2}+(x-\mu )^{2}}}}}\;e^{\delta \gamma +\beta (x-\mu )}}$ ${\displaystyle K_{j}}$ denotes a modified Bessel function of the third kind[1] ${\displaystyle \mu +\delta \beta /\gamma }$ ${\displaystyle \delta \alpha ^{2}/\gamma ^{3}}$ ${\displaystyle 3\beta /(\alpha {\sqrt {\delta \gamma }})}$ ${\displaystyle 3(1+4\beta ^{2}/\alpha ^{2})/(\delta \gamma )}$ ${\displaystyle e^{\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +z)^{2}}})}}$ ${\displaystyle e^{i\mu z+\delta (\gamma -{\sqrt {\alpha ^{2}-(\beta +iz)^{2}}})}}$

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.[2] In the next year Barndorff-Nielsen published the NIG in another paper.[3] It was introduced in the mathematical 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]

## Properties

### Moments

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

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

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

### Convolution

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

## Related distributions

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

## Stochastic process

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.

## As a variance-mean mixture

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]

## References

1. ^ Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 Note: in the literature this function is also referred to as Modified Bessel function of the third kind
2. ^ Barndorff-Nielsen, Ole (1977). "Exponentially decreasing distributions for the logarithm of particle size". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. The Royal Society. 353 (1674): 401–409. doi:10.1098/rspa.1977.0041. JSTOR 79167.
3. ^ O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978
4. ^ O. Barndorff-Nielsen, Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 1997
5. ^ S.T Rachev, Handbook of Heavy Tailed Distributions in Finance, Volume 1: Handbooks in Finance, Book 1, North Holland 2003
6. ^ Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium
7. ^ Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007
8. ^ Paolella, Marc S (2007). Intermediate Probability: A computational Approach. John Wiley & Sons.
9. ^ Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013
10. ^ Karlis, Dimitris (2002). "An EM Type Algorithm for ML estimation for the Normal–Inverse Gaussian Distribution". Statistics and Probability Letters. 57: 43–52.