Nonparametric estimation of conditional quantiles using quantile regression trees

A nonparametric regression method that blends key features of piecewise polynomial quantile regression and tree-structured regression based on adaptive recursive partitioning of the covariate space is investigated. Unlike least-squares regression trees, which concentrate on modelling the relationship between the response and the covariates at the Centre of the response distribution, our quantile regression trees can provide insight into the nature of that relationship at the Centre as well as the tails of the response distribution. Our nonparametric regression quantiles have piecewise polynomial forms, where each piece is obtained by fitting a polynomial quantile regression model to the data in a terminal node of a binary decision tree. The decision tree is constructed by recursively partitioning the data based on repeated analyses of the residuals obtained after model fitting with quantile regression. One advantage of the tree structure is that it provides a simple summary of the interactions among the covariates. The asymptotic behaviour of piecewise polynomial quantile regression estimates and the associated derivative estimates are studied under appropriate regularity conditions. The methodology is illustrated with an example on the incidence rates of mumps in the United States.