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]
This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property. If
then[8]
Convolution
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:[9] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and , respectively, then is NIG-distributed with parameters and
The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, arises as a special case by setting and letting .
Let denote the inverse Gaussian distribution and denote the normal distribution. Let , where ; and let , then follows the NIG distribution, with parameters, . 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]
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.