The Process Induced by Slope Components of \(\alpha \) -Regression Quantile

We consider the linear regression model, along with the process induced by its \(\alpha \) -regression quantile, \(0<\alpha <1\) . While only the intercept component of the \(\alpha \) -regression quantile estimates the quantile \(F^{-1}(\alpha )\) of the model errors, the \(\alpha \) also affects the slope components, whose dispersion infinitely increases as \(\alpha \rightarrow 0,1\) , in the same rate as the variance of the sample \(\alpha \) -quantile. The process of the slope components of \(\alpha \) -regression quantile over \(\alpha \in (0,1)\) is asymptotically equivalent to the process of R-estimates of the slope parameters in the linear model, generated by the Hájek rank scores. Both processes converge to the vector of independent Brownian bridges under exponentially tailed parent distribution F, after standardization by \(f(F ^{-1}(\alpha )).\)