Non-crossing quantile regressions by SVM

Most regression studies focus on the conditional mean estimation. A more informative description of the conditional distribution can be obtained through the conditional quantile estimation. In this paper, we study on nonparametric conditional quantile estimator using support vector (SV) regression approach. We show that a slight modification of Vapnik's epsilon-insensitive SV regression leads to a nonparametric conditional quantile estimator with L-2 regularization. With the great flexibility in nonparametric approach, it is quite possible that two or more estimated conditional quantile functions at different orders can cross or overlap each other. This embarrassing phenomenon is called quantile crossing and it has long been one of the challenging problems in the literature. In this paper, we address the quantile-crossing problem using SV regression approach. With the common use of kernel trick, we derive a non-crossing conditional quantile estimator in the form of a constrained maximization of a piecewise quadratic function. We also propose its efficient Platt's SMO like implementation by exploiting the specific property of the problem.