Normal Inverse Gaussian Distribution
The normal-inverse Gaussian distribution (NIG distribution) is the normal variance-mean mixture with an inverse Gaussian distribution as the mixing density.
The distribution has a location parameter, a scale parameter, and two shape parameters. Its probability density function (PDF) is:
$$f(x) = \frac{\alpha\deltaK_1\left(\alpha\sqrt{\delta^2+(x-\mu)^2}\right)} {\pi\sqrt{\delta^2+(x-\mu)^2}} \exp\left(\delta\sqrt{\alpha^2-\beta^2}+\beta(x-\mu)\right)$$where μ and δ are the location and scale parameters, respectively. In some alternative parameterizations, a factor δ is absorbed into the shape parameters.
The normal inverse Gaussian distribution is implemented by the NormalInverseGaussianDistribution class. It has one constructor that takes the 4 parameters in the following order: location, tail heaviness, asymmetry, and scale. The following constructs a hyperbolic distribution with location 0, scale 1, α = 1.5 and β = 0.8:
var nig = new NormalInverseGaussianDistribution(6.8, 4.1, 1.3, 3.2);Note that evaluation if the Cumulative Distribution Function (CDF) of a hyperbolic distribution is very expensive because no closed form exists and it has to be evaluated using numerical integration of the Probability Density Function. Evaluating the inverse CDF is also expensive.
The NormalInverseGaussianDistribution class has four specific properties that correspond to the parameters of the distribution. The LocationParameter and ScaleParameter properties return the location and scale parameters, respectively. The Alpha and Beta properties return the shape parameters, α and β.
For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..