  Quantile crossing is a common phenomenon in shape constrained nonparametric
quantile regression. A recent study by Wang et al. (2014) has proposed to
address this problem by imposing non-crossing constraints to convex quantile
regression. However, the non-crossing constraints may violate an intrinsic
quantile property. This paper proposes a penalized convex quantile regression
approach that can circumvent quantile crossing while better maintaining the
quantile property. A Monte Carlo study demonstrates the superiority of the
proposed penalized approach in addressing the quantile crossing problem.
