logistic, and Student-$t$ distributions, respectively. More recently, other families of distributions have been proposed by using this idea, such as the log Birnbaum-Saunders distribution [@B08], log skew-elliptical distributions [@MG10], log-generalised extreme value distributions [@RD15], and log-scale mixtures of normals [@VS15].

In this paper, we propose a new class of flexible distributions with support on ${\mathbb R}_+$ by applying the second method to the family of two-piece distributions [@FS98; @A05; @RS14]. In Section \[LTPDistributions\], we introduce the proposed class of distributions and show that it contains very flexible members that can capture a wide variety of shapes and tail behaviours. We show that these models can be seen as a subclass of composite models, which are of great interest in finance. The associated hazard functions are non-monotone with either increasing or decreasing right tails. These distributions are easy to implement using the R packages [@R13] ‘twopiece’ and ‘TPSAS’, which are available upon request. In Section \[MaximumLikelihoodEstimation\], we discuss the properties of the maximum likelihood estimators (MLEs) associated to these models. Although a formal study of the asymptotic properties of the proposed models is beyond the scope of this paper, we present a simulation study which reveals that adding a shape parameter, 