Quantile splines with several covariates

We extend univariate regression quantile splines to problems with several covariates. We adopt an ANOVA-type decomposition approach with main effects captured by linear splines and second-order 'interactions' modeled by bi-linear tensor-product splines. Both univariate linear splines and bi-linear tensor-product splines are optimal when fidelity to data are balanced by a roughness penalty on the fitted function. The problem of sub-model selection and asymptotic justification for using a smaller sub-space of the spline functions in the approximation are discussed. Two examples are considered to illustrate the empirical performance of the proposed methods.