Bayesian quantile regression and variable selection for partial linear single-index model: Using free knot spline

Partial linear single-index model (PLSIM) has both the flexibility of nonparametric treatment and interpretability of linear term, yet existing literatures about it mainly focused on mean regression, and quantile regression analysis is scarce. Based on free knot spline approximation, we apply asymmetric Laplace distribution to implement Bayesian quantile regression, and perform variable selection in linear term and index vector via binary indicators. Our approach is exempt from regularity conditions in frequentist method, and could execute variable selection and quantile regression under mutual posterior correction, which is also the first work to implement them jointly for PLSIM in fully Bayesian framework. The numerical simulation manifests the superiority of our approach to previous methods, which embodied in better efficiency of variable selection, index vector estimates and link function approximation with different error distributions. For illustration of its application, we build a power consumption model of A₂/O process in wastewater treatment and emphatically analyze the impact of water quality factors.