density. For example, in the cases where $s(\cdot;\delta)$ is either a Student-$t$ density with $\delta>0$ degrees of freedom or an exponential power density with power parameter $\delta>0$ (see Appendix A).

A real random variable $X$ is said to be distributed according to a two-piece distribution if its probability density function (PDF) is given by (see *e.g.* [@RS14]): $$\begin{aligned}
\label{TP}
s_{tp}(x;\mu,\sigma_1,\sigma_2,\delta) = \dfrac{2}{\sigma_1+\sigma_2}\left[s\left(\dfrac{x-\mu}{\sigma_1};\delta\right) I(x<\mu) + s\left(\dfrac{x-\mu}{\sigma_2};\delta\right) I(x\geq \mu) \right].\end{aligned}$$

This is, a two-piece density is obtained by continuously joining two half-$s$ densities with different scale parameters on either side of the location $\mu$. The density (\[TP\]) is unimodal, with mode at $\mu$, it is asymmetric for $\sigma_1\neq \sigma_2$, and coincides with the original density $s$ for $\sigma_1=\sigma_2$. Moreover, the tail behaviour of the PDF in (\[TP\]) is the same in each direction, by construction. A popular reparameterisation is obtained by redefining $\sigma_1=\sigma a(\gamma)$ and $\sigma_2 = \sigma b(\gamma)$, where $a(\cdot)$ and $b(\cdot)$ are positive functions of the parameter $\gamma$ [@A05]. Two common choices for $a(\cdot)$ and $b(\cdot)$ are the inverse scale factors $\{a(\gamma),b(\gamma)\}=\{\gamma,1/\gamma\}$, $\gamma \in {\mathbb R}_+$ [@FS98], and the epsilon-skew parameterisation $\{a(\gamma),b(\gamma)\}=\{1-\gamma,1+\gamma\}$, $\gamma \in (-1,1)$ [@MH00]. Other parameterisations are explored in [@RS14]. The PDF associated to this reparameterisation is given 