Generalized hyperbolic distribution

Parameters (notation from wikipedia)

• $\lambda$: shape
• $\alpha$: shape
• $\beta$: shape
• $\delta$: scale (???)
• $\mu$: location

PDF: $$\begin{array}{}\text{(1)}& f(x,\lambda ,\alpha ,\beta ,\mu ,\delta )& =\frac{{\left(\frac{\sqrt{{\alpha }^2-{\beta }^2}}{\delta }\right)}^{\lambda }{e}^{\beta (x-\mu )}}{\sqrt{2\pi }{K}_{\lambda }(\delta \sqrt{{\alpha }^2-{\beta }^2})}\frac{K_{\lambda -\frac{1}{2}}\left(\alpha \sqrt{{\delta }^2+(x-\mu {)}^2}\right)}{{\left(\frac{\sqrt{{\delta }^2+(x-\mu {)}^2}}{\alpha }\right)}^{\frac{1}{2}-\lambda }}\text{(2)}& & =\frac{{\left(\frac{\sqrt{{\alpha }^2-{\beta }^2}}{\delta }\right)}^{\lambda }{e}^{\delta \beta \left(\frac{x-\mu }{\delta }\right)}}{\sqrt{2\pi }{K}_{\lambda }(\delta \sqrt{{\alpha }^2-{\beta }^2})}\frac{K_{\lambda -\frac{1}{2}}\left(\delta \alpha \sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}{{\left(\frac{\delta }{\alpha }\sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}^{\frac{1}{2}-\lambda }}\text{(3)}& & =\frac{1}{\delta }\frac{{\left(\frac{\sqrt{(\delta \alpha {)}^2-(\delta \beta {)}^2}}{\delta \alpha }\right)}^{\lambda }(\delta \alpha {)}^{\frac{1}{2}}{e}^{\delta \beta \left(\frac{x-\mu }{\delta }\right)}}{\sqrt{2\pi }{K}_{\lambda }\left(\sqrt{(\delta \alpha {)}^2-(\delta \beta {)}^2}\right)}\frac{K_{\lambda -\frac{1}{2}}\left(\delta \alpha \sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}{{\left(\sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}^{\frac{1}{2}-\lambda }}\end{array}$$ Note $\alpha$ and $\beta$ always occur as a product $\delta \alpha$ and $\delta \beta$, respectively. Let $$\begin{array}{}\text{(4)}& \hat{\alpha }& =\delta \alpha \text{(5)}& \hat{\beta }& =\delta \beta \end{array}$$ Then $$f(x,\lambda ,\hat{\alpha },\hat{\beta },\mu ,\delta )=\frac{1}{\delta }\frac{{\left(\frac{\sqrt{{\hat{\alpha }}^2-{\hat{\beta }}^2}}{\hat{\alpha }}\right)}^{\lambda }{\hat{\alpha }}^{\frac{1}{2}}{e}^{\hat{\beta }\left(\frac{x-\mu }{\delta }\right)}}{\sqrt{2\pi }{K}_{\lambda }\left(\sqrt{{\hat{\alpha }}^2-{\hat{\beta }}^2}\right)}\frac{K_{\lambda -\frac{1}{2}}\left(\hat{\alpha }\sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}{{\left(\sqrt{1+{\left(\frac{x-\mu }{\delta }\right)}^2}\right)}^{\frac{1}{2}-\lambda }}$$ This is now a true location-scale family, with location $\mu$ and scale $\delta$, but the definition of two of the shape parameters is not the same as in the wikipedia article.