c © 2006, Indian Statistical Institute Quantile Regression with Shape-Constrained Varying Coefficients

Although much research has been devoted to shape-constrained function es-timation, the efforts have been practically confined to the case of univariate smoothing where the unknown function is a function of a single variable. We extend shape-constrained function estimation to a general class of con-strained nonparametric or semi-parametric regression where the nonparamet-ric component can be described by one-dimensional smooth functions. Built on the ideas of He and Shi (1998) and He and Ng (1999), we consider quantile regression with shape constrained coefficient functions. B-splines are used to approximate the unknown coefficient functions, and shape constraints are imposed on the spline coefficients. The method can be implemented with any existing linear program and knot selection algorithm. We show that the method does not compromise smoothness of the estimators, flexibility of the model or computational efficiency. Asymptotic results show that the con-strained B-spline estimators have the same rate of convergence and the same normal limiting distribution as the unconstrained estimators. The method can accommodate a general class of linearizable shape constraints such as convexity/concavity, monotonicity, periodicity and pointwise constraints