via two-piece transformations, has little effect on the performance of the maximum likelihood estimators. In Section \[NumericalExamples\], we present two kinds of applications with real data. In the first example, we illustrate the use of the proposed distributions in the context of data fitting. The main application is presented in the second and third examples, where we employ the proposed distributions for modelling the errors in an accelerated failure time model (AFT) with applications to medical data. In the third example, we discuss the use of a certain class of prediction intervals of the remaining life, which are informative for individual prognosis. In all of these examples, we discuss model selection between some appropriate competitors and the selection of the baseline distribution in the proposed family of distributions.

Log Two-piece Distributions {#LTPDistributions}
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For the sake of completeness, let us first recall the definition of two-piece distributions. Let $s(\cdot;\delta)$ be a symmetric unimodal density, with mode at $0$, with support on $\mathbb R$, and let $\delta\in\Delta\subset{\mathbb R}$ be a shape parameter (location and scale parameters can be added in the usual way). The corresponding CDF will be denoted as $S(\cdot;\delta)$. The shape parameter $\delta$ typically controls the tails of the 