Parameters  location (real) tail heaviness (real) asymmetry parameter (real) scale parameter (real) 

Support  
denotes a modified Bessel function of the third kind^{[1]}  
Mean  
Variance  
Skewness  
Ex. kurtosis  
MGF  
CF 
The normalinverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variancemean 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 BarndorffNielsen.^{[2]} In the next year BarndorffNielsen published the NIG in another paper.^{[3]} It was introduced in the mathematical finance literature in 1997.^{[4]}
The parameters of the normalinverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIGtriangle.^{[5]}
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Using Inverse Normal (Normal Distribution 4)

Inverse Normal distribution: in the Natural Exponential Family

Normal distributions and inverse normal

Minimal Sufficient Statistic: Inverse Normal distribution

Inverse Normal Function
Transcription
Contents
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.
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 normalinverse Gaussian distributions is closed under convolution in the following sense:^{[8]} if and are independent random variables that are NIGdistributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and , respectively, then is NIGdistributed with parameters and
Related distributions
The class of NIG distributions is a flexible system of distributions that includes fattailed and skewed distributions, and the normal distribution, arises as a special case by setting and letting .
Stochastic process
The normalinverse Gaussian distribution can also be seen as the marginal distribution of the normalinverse 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 normalinverse Gaussian process is the timechanged process . The process at time 1 has the normalinverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.
References
 ^ Ole E BarndorffNielsen, 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
 ^ BarndorffNielsen, 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.
 ^ O. BarndorffNielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978
 ^ O. BarndorffNielsen, 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
 ^ Ole E BarndorffNielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013