Normal-inverse Gaussian (NIG)
Parameters location (real)
tail heaviness (real)
asymmetry parameter (real)
scale parameter (real)
Support
PDF

denotes a modified Bessel function of the third kind[1]
Mean
Variance
Skewness
Ex. kurtosis
MGF
CF

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]

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

then[8]

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 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

Related distributions

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 .

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), , 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 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 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]

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.