Partially linear modeling of conditional quantiles using penalized splines

We consider the estimation problem of conditional quantile when multi-dimensional covariates are involved. To overcome the “curse of dimensionality” yet retain model flexibility, we propose two partially linear models for conditional quantiles: partially linear single-index models (QPLSIM) and partially linear additive models (QPLAM). The unknown univariate functions are estimated by penalized splines. An approximate iteratively reweighted penalized least square algorithm is developed. To facilitate model comparisons, we develop effective model degrees of freedom for penalized spline conditional quantiles. Two smoothing parameter selection criteria, Generalized Approximate Cross-validation (GACV) and Schwartz-type Information Criterion (SIC) are studied. Some asymptotic properties are established. Finite sample properties are investigated through simulation studies. Application to the Boston Housing data demonstrates the success of the proposed approach. Both simulations and real applications show encouraging results of the proposed estimators.