Process of the slope components of α-regression quantile

06/08/2021

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by   Jana Jurečková, et al.

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We consider the linear regression model along with the process of its α-regression quantile, 0<α<1. We are interested mainly in the slope components of α-regression quantile and in their dependence on the choice of α. While they are invariant to the location, and only the intercept part of the α-regression quantile estimates the quantile F^-1(α) of the model errors, their dispersion depends on α and is infinitely increasing as α→ 0,1, in the same rate as for the ordinary quantiles. We study the process of R-estimators of the slope parameters over α∈[0,1], generated by the Hájek rank scores. We show that this process, standardized by f(F ^-1(α)) 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 α-regression quantile.