Regression Quantile and Averaged Regression Quantile Processes

Abstract

We consider the averaged version $\widetilde{B}_n(\alpha )$ of the two-step regression $\alpha$-quantile, introduced in [6] and studied in [7]. We show that it is asymptotically equivalent to the averaged version $\bar{B}_n(\alpha )$ of ordinary regression quantile and also study the finite-sample relation of $\widetilde{B}_n(\alpha )$ to $\bar{B}_n(\alpha )$. An interest of its own has the fact that the vector of slope components of the regression $\alpha$-quantile coincides with a particular R-estimator of the slope components of regression parameter. Under a finite n,  the stochastic processes $\widetilde{\mathscr {B}}_n=\{\widetilde{B}_n(\alpha ): \; 0<\alpha <1\}$ and $\mathscr {B}_n=\{\bar{B}_n(\alpha ): \; 0<\alpha <1\}$ differ only by a drift.