Bayesian single-index quantile regression: spline link estimation and spike-and-slab variable selection in high dimensions*

The Single-Index Quantile Regression (SIQR) framework reduces dimensionality while capturing nonlinear relationships between predictors and response quantiles. However, existing methods struggle with high-dimensional settings, variable selection, and adaptive estimation of the link function. To address these challenges, we propose a Bayesian Single-Index Quantile Regression (SAS–SIQR) model that integrates spike–and–slab priors for variable selection and spline-based estimation for the link function. Our approach employs a continuous spike–and–slab before inducing sparsity in the index coefficients and a B-spline representation to model nonlinear effects. Through Reversible Jump MCMC (RJMCMC) and Stochastic Search Variable Selection (SSVS), the model simultaneously performs adaptive knot selection and coefficient estimation. Comprehensive simulations demonstrate that SAS–SIQR outperforms classical SIQR and Bayesian Lasso SIQR (BL–SIQR) in variable selection accuracy, predictive performance, and robustness to outliers, particularly at extreme quantiles. Real–data applications to a contaminated benchmark dataset and a genetic risk score analysis confirm its superiority in handling heteroskedasticity and contamination, yielding interpretable and sparse models with reliable uncertainty quantification.