Normal-inverse Gaussian distribution

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Normal-inverse Gaussian (NIG)
Parameters \mu location (real)
\alpha tail heaviness (real)
\beta asymmetry parameter (real)
\delta scale parameter (real)
\gamma = \sqrt{\alpha^2 - \beta^2}
Support x \in (-\infty; +\infty)\!
pdf \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)}

K_j denotes a modified Bessel function of the second kind[1]
Mean \mu + \delta \beta / \gamma
Variance \delta\alpha^2/\gamma^3
Skewness  3 \beta /(\alpha \sqrt{\delta \gamma})
Ex. kurtosis 3(1+4 \beta^2/\alpha^2)/(\delta\gamma)
MGF e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}
CF e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}

The normal-inverse Gaussian distribution (NIG) is 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 introduced by Ole Barndorff-Nielsen[2] and is a subclass of the generalised hyperbolic distribution. 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.

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.[3][4] The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:[5] 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.

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 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

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


  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. ^ Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium
  4. ^ Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007
  5. ^ Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013