Asymptotic distribution of conical-hull estimators of directional edges

TL;DR

This paper investigates the asymptotic properties of conical-hull estimators for directional edges under the constant returns-to-scale assumption, providing theoretical insights and practical tools for improved efficiency analysis.

Contribution

It derives the asymptotic distribution and convergence rate of the conical-hull estimator under CRS, and introduces bias correction and confidence interval construction methods.

Findings

Conical-hull estimator converges faster than VRS estimator.

Asymptotic distribution can be simulated for inference.

Bias correction improves median squared error in simulations.

Abstract

Nonparametric data envelopment analysis (DEA) estimators have been widely applied in analysis of productive efficiency. Typically they are defined in terms of convex-hulls of the observed combinations of $\mathrm{inputs}\times\mathrm{outputs}$ in a sample of enterprises. The shape of the convex-hull relies on a hypothesis on the shape of the technology, defined as the boundary of the set of technically attainable points in the $\mathrm{inputs}\times\mathrm{outputs}$ space. So far, only the statistical properties of the smallest convex polyhedron enveloping the data points has been considered which corresponds to a situation where the technology presents variable returns-to-scale (VRS). This paper analyzes the case where the most common constant returns-to-scale (CRS) hypothesis is assumed. Here the DEA is defined as the smallest conical-hull with vertex at the origin enveloping the cloud of observed points. In this paper we determine the asymptotic properties of this estimator, showing that the rate of convergence is better than for the VRS estimator. We derive also its asymptotic sampling distribution with a practical way to simulate it. This allows to define a bias-corrected estimator and to build confidence intervals for the frontier. We compare in a simulated example the bias-corrected estimator with the original conical-hull estimator and show its superiority in terms of median squared error.