Process of the slope components of $α$-regression quantile

We consider the linear regression model along with the process of its $\alpha$-regression quantile, $0<\alpha<1$. We are interested mainly in the slope components of $\alpha$-regression quantile and in their dependence on the choice of $\alpha.$ While they are invariant to the location, and only the intercept part of the $\alpha$-regression quantile estimates the quantile $F^{-1}(\alpha)$ of the model errors, their dispersion depends on $\alpha$ and is infinitely increasing as $\alpha\rightarrow 0,1$, in the same rate as for the ordinary quantiles. We study the process of $R$-estimators of the slope parameters over $\alpha\in[0,1]$, generated by the H\'{a}jek rank scores. We show that this process, standardized by $f(F ^{-1}(\alpha))$ under exponentially tailed $F$, converges to the vector of independent Brownian bridges. The same course is true for the process of the slope components of $\alpha$-regression quantile.