Hyperbolic Distribution
The hyperbolic distribution is a distribution whose probability density function (PDF) is the logarithm of a hyperbola. It has heavier tails than the normal distribution. It can be used to model financial assets and turbulent wind speeds.
The Hyperbolic distribution has a location parameter, a scale parameter, and two shape parameters, usually denoted by the Greek letters α (tail heaviness) and β (asymmetry). Its probability density function (PDF) is:
$$f(x) = \frac{\gamma}{2\alpha\delta K_1(\delta\gamma)}e^{-\alpha\sqrt{\delta^2+(x-\mu)^2}+\beta(x-\mu)}$$where μ and δ are the location and scale parameters, respectively, and
$$\gamma = \sqrt{\alpha^2-\beta^2}.$$The hyperbolic distribution is implemented by the HyperbolicDistribution 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 hyperbolic = new HyperbolicDistribution(0, 1.5, 0.8, 1);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 HyperbolicDistribution class has four specific properties that correspond to the parameters of the distribution. The TailHeavinessParameter and AsymmetryParameter properties return the shape parameters, α and β. The LocationParameter and ScaleParameter properties return the location and scale parameters, respectively.
For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..