Non-crossing non-parametric estimates of quantile curves

Since the introduction by Koenker and Bassett, quantile regression has become increasingly important in many applications. However, many non-parametric conditional quantile estimates yield crossing quantile curves (calculated for various p is an element of (0, 1)). We propose a new non-parametric estimate of conditional quantiles that avoids this problem. The method uses an initial estimate of the conditional distribution function in the first step and solves the problem of inversion and monotonization with respect to p is an element of (0, 1) simultaneously. It is demonstrated that the new estimates are asymptotically normally distributed with the same asymptotic bias and variance as quantile estimates that are obtained by inversion of a locally constant or locally linear smoothed conditional distribution function. The performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the method proposed are presented.