Regression Quantile and Averaged Regression Quantile Processes

We consider the averaged version (B) over tilde (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 (B) over bar (n)(alpha) of ordinary regression quantile and also study the finite-sample relation of (B) over tilde (n)(alpha) to (B) over tilde (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 (B) over tilde (n) = {(B) over tilde (n)(alpha) : 0 < alpha 1} and (B) over tilde (n) = {(B) over bar (n)(alpha) : 0 < alpha < 1} differ only by a drift.