Quantile Curves without Crossing

Abstract Conventional regression techniques focus on the conditional averages, but often of more interest are lower or upper conditional quantiles. A more informative description of the relationship among variables can be obtained through regression quantiles (RQ). The percentile curves are usually computed one level at a time. Associated with great flexibility is the embarrassing phenomenon of quantile crossing. We propose a restricted version of regression quantiles (RRQ) that avoids the occurrence of crossing while maintaining sufficient modeling flexibility. RRQ remains in the general framework of the regression quantiles both conceptually and computationally. Because it relates all quantile functions through the conditional median, it also means substantial savings in computation costs when multiple quantiles for high-dimensional data are needed. Two examples in connection with the physical and engineering sciences are presented to demonstrate the usefulness of RRQ curves.