A family of smooth piecewise-linear models with probabilistic interpretations

The smooth piecewise-linear models cover a wide range of applications nowadays. Basically, there are two classes of them: models are transitional or hyperbolic according to their behaviour at the phase-transition zones. This study explored three different approaches to build smooth piecewise-linear models, and we analysed their inter-relationships by a unifying modelling framework. We conceived the smoothed phase-transition zones as domains where a mixture process takes place, which ensured probabilistic interpretations for both hyperbolic and transitional models in the light of random thresholds. Many popular models found in the literature are special cases of our methodology. Furthermore, this study introduces novel regression models as alternatives, such as the Epanechnikov, Normal and Skewed-Normal Bent-Cables.