Normal-inverse Gaussian distribution

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]

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

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.