Weak Convergence of Weighted Empirical Cumulative Quantile Regression Functions

Let （X, Y） be a bivariate random variable and F（x） the marginal distribution function of X. We define h（u） = E（Y|F（X） = u） as the quantile regression （QR） function of Y on X. The Cumulative QR function S（u） with weighted coefficients is defined as the integral of J（·）h（·） over the range O,u, where J（·） is the weight function. In this paper, we discuss the convergence properties of its empirical versions.