Normal-inverse Gaussian distribution

Normal-inverse Gaussian (NIG)

Parameters	[math]\displaystyle{ \mu }[/math] location (real) [math]\displaystyle{ \alpha }[/math] tail heaviness (real) [math]\displaystyle{ \beta }[/math] asymmetry parameter (real) [math]\displaystyle{ \delta }[/math] scale parameter (real) [math]\displaystyle{ \gamma = \sqrt{\alpha^2 - \beta^2} }[/math]
Support	[math]\displaystyle{ x \in (-\infty; +\infty)\! }[/math]
PDF	[math]\displaystyle{ \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)} }[/math] [math]\displaystyle{ K_j }[/math] denotes a modified Bessel function of the second kind
Mean	[math]\displaystyle{ \mu + \delta \beta / \gamma }[/math]
Variance	[math]\displaystyle{ \delta\alpha^2/\gamma^3 }[/math]
Skewness	[math]\displaystyle{ 3 \beta /(\alpha \sqrt{\delta \gamma}) }[/math]
Kurtosis	[math]\displaystyle{ 3(1+4 \beta^2/\alpha^2)/(\delta\gamma) }[/math]
MGF	[math]\displaystyle{ e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})} }[/math]
CF	[math]\displaystyle{ e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})} }[/math]

The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) 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

[math]\displaystyle{ x\sim\mathcal{NIG}(\alpha,\beta,\delta,\mu) \text{ and } y=ax+b, }[/math]

then

[math]\displaystyle{ y\sim\mathcal{NIG}\bigl(\frac{\alpha}{\left|a\right|},\frac{\beta}{a},\left|a\right|\delta,a\mu+b\bigr). }[/math]

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 [math]\displaystyle{ X_1 }[/math] and [math]\displaystyle{ X_2 }[/math] are independent random variables that are NIG-distributed with the same values of the parameters [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math], but possibly different values of the location and scale parameters, [math]\displaystyle{ \mu_1 }[/math], [math]\displaystyle{ \delta_1 }[/math] and [math]\displaystyle{ \mu_2, }[/math] [math]\displaystyle{ \delta_2 }[/math], respectively, then [math]\displaystyle{ X_1 + X_2 }[/math] is NIG-distributed with parameters [math]\displaystyle{ \alpha, }[/math] [math]\displaystyle{ \beta, }[/math][math]\displaystyle{ \mu_1+\mu_2 }[/math] and [math]\displaystyle{ \delta_1 + \delta_2. }[/math]

Related distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, [math]\displaystyle{ N(\mu,\sigma^2), }[/math] arises as a special case by setting [math]\displaystyle{ \beta=0, \delta=\sigma^2\alpha, }[/math] and letting [math]\displaystyle{ \alpha\rightarrow\infty }[/math].

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), [math]\displaystyle{ W^{(\gamma)}(t)=W(t)+\gamma t }[/math], we can define the inverse Gaussian process [math]\displaystyle{ A_t=\inf\{s\gt 0:W^{(\gamma)}(s)=\delta t\}. }[/math] Then given a second independent drifting Brownian motion, [math]\displaystyle{ W^{(\beta)}(t)=\tilde W(t)+\beta t }[/math], the normal-inverse Gaussian process is the time-changed process [math]\displaystyle{ X_t=W^{(\beta)}(A_t) }[/math]. The process [math]\displaystyle{ X(t) }[/math] at time [math]\displaystyle{ t=1 }[/math] 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 [math]\displaystyle{ \mathcal{IG} }[/math] denote the inverse Gaussian distribution and [math]\displaystyle{ \mathcal{N} }[/math] denote the normal distribution. Let [math]\displaystyle{ z\sim\mathcal{IG}(\delta,\gamma) }[/math], where [math]\displaystyle{ \gamma=\sqrt{\alpha^2-\beta^2} }[/math]; and let [math]\displaystyle{ x\sim\mathcal{N}(\mu+\beta z,z) }[/math], then [math]\displaystyle{ x }[/math] follows the NIG distribution, with parameters, [math]\displaystyle{ \alpha,\beta,\delta,\mu }[/math]. 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.