The SST distribution is a reparametrization of $ST3$, which is the skew t-student type 3. Below is some information.
Let $Z_0 \sim ST3 (0, 1, \nu, \tau)$ and $Y = \mu + \sigma\left(\frac{Z_0 - m}{s}\right)$, where,
\begin{equation} m = E(Z_0) = \frac{2\tau^\frac{1}{2}(\nu-\nu^{-1})}{(\tau - 1)B\left(\frac{1}{2},\frac{\tau}{2}\right)} \end{equation}
\begin{equation} s^2 = Var(Z_0) = \frac{\tau}{\tau - 2}(\nu^2 + \nu^{-2} - 1) - m^2 \end{equation}
Hence, $Y = \mu_0 + \sigma_0Z_0$, where $\mu_0 = \mu - \frac{\sigma m}{s}$ e $\sigma_0 = \frac{\sigma}{s}$, and so $Y \sim ST3(\mu_0, \sigma_0, \nu, \tau)$ with $E(Y) = \mu$ and $Var(Y) = \sigma^2$ for $\tau > 2$. Let $Y \sim SST(\mu, \sigma, \nu, \tau) = ST3(\mu_0, \sigma_0, \nu, \tau)$ for $\tau > 2$. The pdf of the skew Student t distribution, denoted by $Y \sim SST(\mu, \sigma, \nu, \tau)$, is given by:
\begin{equation} f_Y(y|\mu, \sigma, \nu, \tau) = \left\{ \begin{array}{rcl} \frac{c}{\sigma_0}\left(1 + \frac{\nu^2 z^2}{\tau} \right)^{\frac{-(\tau + 1)}{2}} & \mbox{if} & y<\mu_0 \\ \frac{c}{\sigma_0}\left(1 + \frac{z^2}{\nu^2 \tau} \right)^{\frac{-(\tau + 1)}{2}} & \mbox{if} & y\geq \mu_0 \end{array}\right. \end{equation}
for $-\infty < y < \infty$, where $-\infty < \mu < \infty$, $\sigma >0$, $\nu > 0$ and $\tau > 2$ and where $z=\frac{(y-\mu_0)}{\sigma_0}$, $\mu_0 = \mu - \frac{\sigma m}{s}$, $\sigma_0 = \frac{\sigma}{s}$ and $c = 2\nu [(1+\nu^2)B\left(\frac{1}{2},\frac{\tau}{2}\right)\tau^{\frac{1}{2}}]^{-1}$.
Note that $E(Y) = \mu$ and $Var(Y) = \sigma^2$ and the moment based skewness and excess kurtosis of $SST(\mu, \sigma, \nu, \tau)$ are the same as for $ST3(\mu_0, \sigma_0, \nu, \tau)$, and hence the same as for $ST3(0, 1, \nu, \tau)$, depending only on $\nu$ and $\tau$.
How would the likelihood function of this distribution be expressed, according to the parameters? It is known that the link functions for $(\mu, \sigma, \nu, \tau)$ is the $identity$, $log$, $log$, and $log-2$.