  Quantile regression and partial frontier are two distinct approaches to
nonparametric quantile frontier estimation. In this article, we demonstrate
that partial frontiers are not quantiles. Both convex and nonconvex
technologies are considered. To this end, we propose convexified order-$\alpha$
as an alternative to convex quantile regression (CQR) and convex expectile
regression (CER), and two new nonconvex estimators: isotonic CQR and isotonic
CER as alternatives to order-$\alpha$. A Monte Carlo study shows that the
partial frontier estimators perform relatively poorly and even can violate the
quantile property, particularly at low quantiles. In addition, the simulation
evidence shows that the indirect expectile approach to estimating quantiles
generally outperforms the direct quantile estimations. We further find that the
convex estimators outperform their nonconvex counterparts owing to their global
shape constraints. An illustration of those estimators is provided using a
real-world dataset of U.S. electric power plants.
