Asymptotic inference for the constrained quantile regression process

I investigate the asymptotic distribution of linear quantile regression coefficient estimates when the parameter lies on the boundary of the parameter space. In order to allow for inferences made across many conditional quantiles, I provide a uniform characterization of constrained quantile regression estimates as a stochastic process over an interval of quantile levels. To do this I pose the process of estimates as solutions to a parameterized family of constrained optimization problems, parameterized by quantile level. A uniform characterization of the dual solution to these problems – the so-called regression rankscore process – is also derived, which can be used for score-type inference in quantile regression. The asymptotic behavior of quasi-likelihood ratio, Wald and regression rankscore processes for inference when the null hypothesis asserts that the parameters lie on a boundary follows from the features of the constrained solutions.