by $$\begin{aligned}
\label{TP2}
s_{tp}(x;\mu,\sigma,\gamma,\delta) = \dfrac{2}{\sigma[a(\gamma)+b(\gamma)]}\left[s\left(\dfrac{x-\mu}{\sigma b(\gamma)};\delta\right) I(x<\mu) + s\left(\dfrac{x-\mu}{\sigma a(\gamma)};\delta\right) I(x\geq \mu) \right]\end{aligned}$$ This transformation preserves the existence of moments and the ease of use of the original distribution $s$. The corresponding cumulative distribution function and quantile function can be easily obtained from this expression (see [@A05]). This class of distributions has been shown to have good inferential properties for regular choices of the baseline density $s$ [@A05; @JA10].

By applying method (ii), described in Section \[sec:intro\], to the family of two-piece distributions, we can produce distributions with support on ${\mathbb R}_+$ as follows.

A positive random variable $Y$ is said to be distributed according to a log two-piece (LTP) distribution if its PDF is given by: $$\begin{aligned}
\label{LTPPDF}
s_l(y; \mu,\sigma,\gamma,\delta) = \dfrac{2}{y\sigma[a(\gamma)+b(\gamma)]} &\left[ s\left(\dfrac{\log(y)-\mu}{\sigma b(\gamma)};\delta\right)I(y<e^{\mu})\right.\nonumber\\ &\quad\quad+\left.s\left(\dfrac{\log(y)-\mu}{\sigma a(\gamma)};\delta\right)I(y\geq e^{\mu}) \right].\end{aligned}$$

Given that the class of two-piece distributions contains all the symmetric unimodal distributions with support on the real line, it follows that the class of LTP distributions contains the class of log-symmetric distributions as well as models such that the distribution of $\log Y$ is asymmetric. The LTP Laplace distribution, which is obtained by using a Laplace baseline density $s$ in (\[LTPPDF\]), has been studied in [@K01]. However, other 