Unified Non-crossing Multiple Quantile Regressions Tree

In this paper, we consider the estimation problem of a tree model for multiple conditional quantile functions of the response. Using the GUIDE algorithm, the quantile regression tree (QRT) method has been developed to construct a tree model for an individual quantile function. However, QRT produces different tree models across quantile levels because it estimates several quantile regression tree models separately. Furthermore, the estimated quantile functions from QRT often cross each other and consequently violate the basic properties of quantiles. This undesirable phenomenon reduces prediction accuracy and makes it difficult to interpret the resulting tree models. To overcome such limitations, we propose the unified non-crossing multiple quantile regressions tree (UNQRT) method, which constructs a common tree structure across all interesting quantile levels for better data visualization and model interpretation. Furthermore, the UNQRT estimates non-crossing multiple quantile functions simultaneously by enforcing non-crossing constraints, resulting in the improvement of prediction accuracy. The numerical results are presented to demonstrate the competitive performance of the proposed UNQRT over QRT.