Normal-inverse Gaussian distribution
|Parameters|| location (real)
tail heaviness (real)
asymmetry parameter (real)
scale parameter (real)
denotes a modified Bessel function of the second kind
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 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, arises as a special case by setting and letting .
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: 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 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), , we can define the inverse Gaussian process Then given a second independent drifting Brownian motion, , the normal-inverse Gaussian process is the time-changed process . The process 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.
- 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 (The Royal Society) 353 (1674): 401–409. doi:10.1098/rspa.1977.0041. JSTOR 79167.
- 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
- Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013