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# Normal-inverse Gaussian distribution

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 Title: Normal-inverse Gaussian distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Normal-inverse Gaussian distribution

 Parameters \mu location (real) \alpha tail heaviness (real) \beta asymmetry parameter (real) \delta scale parameter (real) \gamma = \sqrt{\alpha^2 - \beta^2} x \in (-\infty; +\infty)\! \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 third kind \mu + \delta \beta / \gamma \delta\alpha^2/\gamma^3 3 \beta /(\alpha \sqrt{\delta \gamma}) 3(1+4 \beta^2/\alpha^2)/(\delta\gamma) e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})} 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 noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen, in the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.

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

## Contents

• Properties 1
• Moments 1.1
• Linear transformation 1.2
• Summation 1.3
• Convolution 1.4
• Related Distributions 2
• Stochastic Process 3
• References 4

## Properties

### Moments

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.

### 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 normal-inverse Gaussian distributions is closed under convolution in the following sense: 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.

## Related Distributions

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.

## 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), 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.