Asymptotic Properties of some Non-parametric Hyperbolic Efficiency Estimators

A hyperbolic measure of technical efficiency was proposed by Färe et al. (1985) wherein efficiency is measured by the simultaneous maximum, feasible reduction in input quantities and increase in output quantities. In cases where returns to scale are not constant, the non-parametric data envelopment analysis (DEA) estimator of hyperbolic efficiency cannot be written as a linear program; consequently, the measure has not been used in empirical studies except where returns to scale are constant, allowing the estimator to be computed by linear programming methods. This paper develops an alternative estimator of the hyperbolic measure proposed by Färe et al.. Statistical consistency and rates of convergence are established for the new estimator. A numerical procedure allowing computation of the original estimator is provided, and this estimator is also shown to be consistent, with the same rate of convergence as the new estimator. In addition, an unconditional, hyperbolic order-m efficiency estimator is developed by extending the ideas of Cazals et al. (2002). Asymptotic properties of this estimator are also given. ∗The John E. Walker Department of Economics, 222 Sirrine Hall, Clemson University, Clemson, South Carolina 29634–1309, USA; email [email protected]. This work was made possible by the Palmetto cluster operated and maintained by the Clemson Computing and Information Technology group at Clemson University. In addition, I have benefited from numerous discussions with Léopold Simar and other members of the Institut de Statistique, Université Catholique de Louvain in Louvain-la-Neuve over the years. Any errors are solely my responsibility.