\prod_{j\in\text{Int}} \left[ S_l(T_j^R;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)-S_l(T_j^L;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)\right],\end{aligned}$$

with the notation discussed previously. It is important to notice that by using asymmetric errors, we obtain a curve that does not represent the mean response. However, as discussed in [@AG08], this lack of centring can be calibrated after estimating the parameters by adding a suitable quantity $M_{\varepsilon}$ which reflects the lack of centring of the errors. For instance, in order to obtain the mean response, we can use $M_{\varepsilon}=-{\mathbb E}[\varepsilon_j]$, computed at the MLE of the parameters of the error distribution. This strategy will only affect the intercept parameter. When using baseline models with infinite variance (such as a log-Cauchy distribution), one might opt for centring around the median (or another quantile), instead of the mean. A formal study of the asymptotic properties of the MLEs under different types of censoring is beyond the scope of this paper. However, in Section \[SimulationStudy\], we illustrate the performance of the MLEs in a linear regression model with censored observations through a simulation study.

Simulation Study {#SimulationStudy}
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In this section, we present a simulation study in order to illustrate the performance of the MLEs of the parameters of some LTP distributions. Throughout, we employ the epsilon-skew parameterisation discussed 