On monotonicity of regression quantile functions

In the linear regression quantile model, the conditional quantile of the response, YY, given xx is QY|x(τ)≡x′β(τ)QY|x(τ)≡x′β(τ). Though QY|x(τ)QY|x(τ) must be monotonically increasing, the Koenker–Bassett regression quantile estimator, x′βˆ(τ), is not monotonic outside a vanishingly small neighborhood of x=x̄. Given a grid of mesh δnδn, let x′βˆ∗(τ) be the linear interpolation of the values of x′βˆ(τ) along the grid. We show here that for a range of rates, δnδn, x′βˆ∗(τ) will be strictly monotonic (with probability tending to one) and will be asymptotically equivalent to x′βˆ(τ) in the sense that n1/2n1/2 times the difference tends to zero at a rate depending on δnδn.