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The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) 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

x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,

then^[8]

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

Related distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, $N(\mu ,\sigma ^{2}),$ arises as a special case by setting $\beta =0,\delta =\sigma ^{2}\alpha ,$ and letting $\alpha \rightarrow \infty$ .

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), $W^{(\gamma )}(t)=W(t)+\gamma t$ , 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 $t=1$ 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 ${\mathcal {IG}}$ denote the inverse Gaussian distribution and ${\mathcal {N}}$ denote the normal distribution. Let $z\sim {\mathcal {IG}}(\delta ,\gamma )$ , where $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}$ ; and let $x\sim {\mathcal {N}}(\mu +\beta z,z)$ , then $x$ follows the NIG distribution, with parameters, $\alpha ,\beta ,\delta ,\mu$ . 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

^ 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
^ 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. 353 (1674). The Royal Society: 401–409. doi:10.1098/rspa.1977.0041. JSTOR 79167.
^ O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978
^ O. Barndorff-Nielsen, Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 1997
^ S.T Rachev, Handbook of Heavy Tailed Distributions in Finance, Volume 1: Handbooks in Finance, Book 1, North Holland 2003
^ Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium
^ Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007
^ Paolella, Marc S (2007). Intermediate Probability: A computational Approach. John Wiley & Sons.
^ Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013
^ Karlis, Dimitris (2002). "An EM Type Algorithm for ML estimation for the Normal–Inverse Gaussian Distribution". Statistics and Probability Letters. 57: 43–52.

[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. 353 (1674). The Royal Society: 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.

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