Generalized quantile and expectile properties for shape constrained nonparametric estimation

Convex quantile regression (CQR) is a fully nonparametric approach to estimating quantile functions, which has proved useful in many applications of productivity and efficiency analysis. Importantly, CQR satisfies the quantile property, which states that the observed data is split into proportions by the CQR frontier for any weight in the unit interval. Convex expectile regression (CER) is a closely related nonparametric approach, which has the following expectile property: the relative share of negative deviations is equal to the weight of negative deviations. The first contribution of this paper is to extend these quantile and expectile properties to the general set of shape constrained nonparametric functions. The second contribution is to relax the global concavity assumptions of the CQR and CER estimators, developing the isotonic nonparametric quantile and expectile estimators. Our third contribution is to compare the finite sample performance of the CQR and CER approaches in the controlled environment of Monte Carlo simulations.