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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.

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  • $\begingroup$ No, they are not the same. The difference is the way skewness is introduced. Hansen's skew-$t$ is a 'two-piece' distribution in the spirit of the transformation in this paper. This transformation preserves the tails. The special case of the Generalised Hyperbolic Distribution has different tails each side of the mode, reference. There are many many skew-$t$ distributions. $\endgroup$ – user10525 May 1 '12 at 18:17
  • $\begingroup$ Thank you @Procrastinator! The paper of Aas & Haff(2006) you posted gives a detailed explanation. Do you know if the sstd, which stands for skewed standardised t distribution, in R refers to the latter (the special case of GHD)? $\endgroup$ – onethird May 1 '12 at 22:54
  • $\begingroup$ If you refer to this package, then I think dsstd is neither of these. It is used for the 'two-piece' t with the parameterisation in Fernandez and Steel (1998) which is referred as to 'inverse scale factors'. $\endgroup$ – user10525 May 1 '12 at 23:08
  • $\begingroup$ These two distribution are quite different from one another. The pdf of the skewed $t$-distribution, the limiting case of the generalized hyperbolic distribution, is expressible in terms of modified Bessel function of the second kind $\endgroup$ – Sasha Aug 21 '13 at 16:44
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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.

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