Noncrossing quantile regression curve estimation

Abstract

Since quantile regression curves are estimated individually, the quantile curves can cross, leading to an invalid distribution for the response. A simple constrained version of quantile regression is proposed to avoid the crossing problem for both linear and nonparametric quantile curves. A simulation study and a reanalysis of tropical cyclone intensity data shows the usefulness of the procedure. Asymptotic properties of the estimator are equivalent to the typical approach under standard conditions, and the proposed estimator reduces to the classical one if there is no crossing. The performance of the constrained estimator has shown significant improvement by adding smoothing and stability across the quantile levels.

Fig. 1

Plot of the true conditional quantile functions for Example 4 (a) and Example 5 (b). The five curves from bottom to top represent τ = 0.1, 0.3, 0.5, 0.7, 0.9.

Fig. 2

Plot of mean squared error in estimation of the slope at the median as a function of the number of included quantiles, for sample sizes n = 50 (solid line), n = 100 (dashed line), and n = 200 (dotted line). Each curve is scaled so that the mean squared error is reported as a ratio relative to that of using only median regression.

Fig. 3

Plot of the estimated slope coefficients at the median (a) and the 0.99 quantile (b) as the number of included quantiles is increased. NAO, North Atlantic Oscillation Index (solid line); SOI, Southern Oscillation Index (dashed line); SST, Atlantic sea surface temperature (dotted line); SUN, average sunspot number (dotted/dashed line).