A unification approach in semi-parametric piecewise models

Piecewise models are powerful tools in statistical analysis, often providing a better fit to data patterns than traditional models. The most commonly used piecewise model, the piecewise exponential model, assumes an exponential distribution for the time interval between change points. However, the assumption of a constant hazard rate between change points may not be realistic in many cases. In this paper, we present a unified approach that introduces a general structure for constructing piecewise models, allowing for different behaviors between change points. The proposed structure yields models that are easier to comprehend and interpret than nonparametric approaches, while providing greater accuracy and flexibility than parametric models. We discuss in detail the mathematical properties of the proposed approach and its application to various baseline models. We first apply a profile likelihood approach to estimate the change points, followed by a maximum likelihood inference to estimate the risk parameters of the model. Additionally, we provide an example application involving the survival times of patients with skin cancer, segmented by sex, to illustrate the effectiveness of the proposed approach.