Parameters λ {displaystyle lambda } (real) α {displaystyle alpha } (real) β {displaystyle eta } asymmetry parameter (real) δ {displaystyle delta } scale parameter (real) μ {displaystyle mu } location (real) γ = α 2 − β 2 {displaystyle gamma ={sqrt {alpha ^{2}-eta ^{2}}}} Support x ∈ ( − ∞ ; + ∞ ) {displaystyle xin (-infty ;+infty )!} PDF ( γ / δ ) λ 2 π K λ ( δ γ ) e β ( x − μ ) {displaystyle {rac {(gamma /delta )^{lambda }}{{sqrt {2pi }}K_{lambda }(delta gamma )}};e^{eta (x-mu )}!} × K λ − 1 / 2 ( α δ 2 + ( x − μ ) 2 ) ( δ 2 + ( x − μ ) 2 / α ) 1 / 2 − λ {displaystyle imes {rac {K_{lambda -1/2}left(alpha {sqrt {delta ^{2}+(x-mu )^{2}}}ight)}{left({sqrt {delta ^{2}+(x-mu )^{2}}}/alpha ight)^{1/2-lambda }}}!} Mean μ + δ β K λ + 1 ( δ γ ) γ K λ ( δ γ ) {displaystyle mu +{rac {delta eta K_{lambda +1}(delta gamma )}{gamma K_{lambda }(delta gamma )}}} Variance δ K λ + 1 ( δ γ ) γ K λ ( δ γ ) + β 2 δ 2 γ 2 ( K λ + 2 ( δ γ ) K λ ( δ γ ) − K λ + 1 2 ( δ γ ) K λ 2 ( δ γ ) ) {displaystyle {rac {delta K_{lambda +1}(delta gamma )}{gamma K_{lambda }(delta gamma )}}+{rac {eta ^{2}delta ^{2}}{gamma ^{2}}}left({rac {K_{lambda +2}(delta gamma )}{K_{lambda }(delta gamma )}}-{rac {K_{lambda +1}^{2}(delta gamma )}{K_{lambda }^{2}(delta gamma )}}ight)} MGF e μ z γ λ ( α 2 − ( β + z ) 2 ) λ K λ ( δ α 2 − ( β + z ) 2 ) K λ ( δ γ ) {displaystyle {rac {e^{mu z}gamma ^{lambda }}{({sqrt {alpha ^{2}-(eta +z)^{2}}})^{lambda }}}{rac {K_{lambda }(delta {sqrt {alpha ^{2}-(eta +z)^{2}}})}{K_{lambda }(delta gamma )}}} |
The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution. Its probability density function (see the box) is given in terms of modified Bessel function of the second kind, denoted by
Contents
Linear transformation
This class is closed under affine transformations.
Summation
Barndorff-Nielsen and Halgreen proved that the GIG distribution has Infinite divisibility and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the GIG distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinite divisible as well.
Related distributions
As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the Laplace distribution, the hyperbolic distribution, the normal-inverse Gaussian distribution and the variance-gamma distribution.
Applications
It is mainly applied to areas that require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails—a property the normal distribution does not possess. The generalised hyperbolic distribution is often used in economics, with particular application in the fields of modelling financial markets and risk management, due to its semi-heavy tails.