Partial frontiers are not quantiles

TL;DR

This paper clarifies that partial frontiers are not equivalent to quantiles and introduces new estimators, demonstrating through simulations and real data that these methods outperform traditional partial frontier approaches.

Contribution

It proposes convexified order-$eta$ and two nonconvex estimators as alternatives to partial frontiers, highlighting their advantages over existing methods.

Findings

Partial frontiers can violate quantile properties, especially at low quantiles.

Indirect expectile methods generally outperform direct quantile estimations.

Convex estimators outperform nonconvex ones due to shape constraints.

Abstract

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.