Some asymptotic results on bivariate quantile splines

Nonparametric estimation of conditional quantiles is of fundamental importance in analyzing general regression problems, especially when heteroscedasticity is suspected. Koenker et al. (Biometrika 81, 1994, 673) generalized the classical Koenker-Bassett regression quantiles to provide spline estimators of conditional quantiles when there is a single regressor. Later, He et al. (J. Roy. Statist. Soc. B 60, 1998, 537) generalised this approach to the case of two regressors. Here we consider asymptotic properties of these bivariate quantile splines. A general consistency result is provided for bivariate quantile smoothing splines, and an asymptotic representation yielding asymptotic normality is developed for fixed-knot B-splines. These results provide asymptotic justification for using quantile splines, and the latter results provide a basis for asymptotic inference.