Semiparametric finite mixture of regression models with Bayesian P-splines

Mixture models provide a useful tool to account for unobserved heterogeneity and are at the basis of many model-based clustering methods. To gain additional flexibility, some model parameters can be expressed as functions of concomitant covariates. In this Paper, a semiparametric finite mixture of regression models is defined, with concomitant information assumed to influence both the component weights and the conditional means. In particular, linear predictors are replaced with smooth functions of the covariate considered by resorting to cubic splines. An estimation procedure within the Bayesian paradigm is suggested, where smoothness of the covariate effects is controlled by suitable choices for the prior distributions of the spline coefficients. A data augmentation scheme based on difference random utility models is exploited to describe the mixture weights as functions of the covariate. The performance of the proposed methodology is investigated via simulation experiments and two real-world datasets, one about baseball salaries and the other concerning nitrogen oxide in engine exhaust.