Normal-inverse Gaussian Distribution

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. 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.

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. If
: $x\sim\mathcal(\alpha,\beta,\delta,\mu) \text y=ax+b,$
then
: $y\sim\mathcal\bigl(\frac,\frac,\left, a\\delta,a\mu+b\bigr).$

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), $W^(t)=W(t)+\gamma t$, we can define the inverse Gaussian process $A_t=\inf\.$ Then given a second independent drifting Brownian motion, $W^(t)=\tilde W(t)+\beta t$, the normal-inverse Gaussian process is the time-changed process $X_t=W^(A_t)$. The process $X(t)$ at time $t=1$ 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 $\mathcal$ denote the inverse Gaussian distribution and $\mathcal$ denote the normal distribution. Let $z\sim\mathcal(\delta,\gamma)$, where $\gamma=\sqrt$; and let $x\sim\mathcal(\mu+\beta z,z)$, then $x$ follows the NIG distribution, with parameters, $\alpha,\beta,\delta,\mu$. 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.

References

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Continuous distributions