The Annals of Statistics

Asymptotic distribution of conical-hull estimators of directional edges

Byeong U. Park, Seok-Oh Jeong, and Léopold Simar

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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 inputs×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 inputs × 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.

Article information

Ann. Statist., Volume 38, Number 3 (2010), 1320-1340.

First available in Project Euclid: 8 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Conical-hull asymptotic distribution efficiency data envelopment analysis DEA constant returns-to-scale CRS


Park, Byeong U.; Jeong, Seok-Oh; Simar, Léopold. Asymptotic distribution of conical-hull estimators of directional edges. Ann. Statist. 38 (2010), no. 3, 1320--1340. doi:10.1214/09-AOS746.

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