Process of the slope components of $\alpha$-regression quantile

Abstract

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 $\alpha \to 0,1$, 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 Hájek 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.