On monotonicity of regression quantile functions

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