Generalized hyperbolic distribution

Parameters (notation from wikipedia)

PDF: (1)f(x,λ,α,β,μ,δ)=(α2β2δ)λeβ(xμ)2πKλ(δα2β2)Kλ12(αδ2+(xμ)2)(δ2+(xμ)2α)12λ(2)=(α2β2δ)λeδβ(xμδ)2πKλ(δα2β2)Kλ12(δα1+(xμδ)2)(δα1+(xμδ)2)12λ(3)=1δ((δα)2(δβ)2δα)λ(δα)12eδβ(xμδ)2πKλ((δα)2(δβ)2)Kλ12(δα1+(xμδ)2)(1+(xμδ)2)12λ Note α and β always occur as a product δα and δβ, respectively. Let (4)α^=δα(5)β^=δβ Then f(x,λ,α^,β^,μ,δ)=1δ(α^2β^2α^)λα^12eβ^(xμδ)2πKλ(α^2β^2)Kλ12(α^1+(xμδ)2)(1+(xμδ)2)12λ This is now a true location-scale family, with location μ and scale δ, but the definition of two of the shape parameters is not the same as in the wikipedia article.