Superquantile efficiency frontiers

Several production frontiers with distinct probabilistic characteristics have been proposed, such as full frontier, quantile frontier, and expectile frontier. This paper introduces an alternative called the superquantile frontier, defined as the upper tail average of the output given a fixed setting of inputs. For a probability level, a production unit lies above the level-τ superquantile frontier if its output exceeds the average output of the top (1-τ) × 100% of production units that use the same inputs. The level-τ superquantile frontier exhibits a hinge-like sensitivity to performance: it responds to units in the top (1-τ) × 100% of the distribution, while remaining insensitive to the bottom τ × 100% unit. The superquantile frontier function can be estimated via (generalized) linear superquantile regression. This paper proposes a convex nonparametric multivariate superquantile regression method for estimating the superquantile frontier, requiring only that the frontier function be nondecreasing and concave in the inputs. The estimation involves solving three sequential continuous linear programs. To illustrate the properties of the estimated superquantile frontiers from the proposed convex nonparametric regression, three experiments on synthetic data and one on real data are presented. The paper also explores two applications of the superquantile frontier: its formulation within the stochastic frontier model, and its use in estimating shadow prices.