I recently studied two asymmetric t distribution both with a name of skewed-$t$. I am confused with their differences or are they actually the same?
The first one is introduced by Hansen (1994) with pdf:
$f(x;\nu,\zeta)=\begin{cases}
\begin{array}{cc}
bc\left(1+\frac{1}{\nu-2}\left(\frac{bx+a}{1-\zeta}\right)^{2}\right)^{-\frac{\nu+1}{2}} & \quad,\text{if }\: x<-\frac{a}{b}\\
bc\left(1+\frac{1}{\nu-2}\left(\frac{bx+a}{1+\zeta}\right)^{2}\right)^{-\frac{\nu+1}{2}} & \quad,\text{if }\: x\geq-\frac{a}{b}
\end{array}\end{cases}$
where $a=4\zeta c\frac{\nu-2}{\nu-1}$, $b^{2}=1+3\zeta^{2}-a^{2}$,
$c=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi(\nu-2)}\Gamma(\frac{\nu}{2})}$ and $\nu$ is DoF while $\zeta$ denotes the skewness.
The other one is a special case of Generalized Hyperbolic Distributions $X \thicksim GH_d(\lambda,\chi,\psi,\mu,\Sigma,\zeta)$ when $\lambda=-0.5\nu$, $\chi=\nu$, $\psi=0$
Any suggestion will be appreciated!
Reference
Hansen, B.E. (1994), Autoregressive conditional density estimation, Intern. Econ. Rev., vol. 35, no. 3, 705–730.
Best Answer
I think the dsstd in package fGarch is the Fernandez and Steel (1998) version, BUT shifted and scaled so that mean is the true mean or expected value of the dsstd distribution and sd is the true standard deviation of the dsstd distribution.