Abstract
A hyperbolic measure of technical efficiency was proposed by Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 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 Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 1985). 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. (J. Econometric. 106:1–25, 2002). Asymptotic properties of this estimator are also given.
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Notes
- 1.
- 2.
One can find numerous published applications of DEA to datasets with 50–150 observations and 5 or more dimensions in the input-output space. DEA-based inefficiency estimates from such studies are likely meaningless in a statistical sense due to the curse of dimensionality problem (see Simar and Wilson (2000b) for discussion).
- 3.
Wheelock and Wilson (2008) also derived asymptotic properties for a hyperbolic FDH efficiency estimator.
- 4.
- 5.
Alternatively, one might consider estimating the directional distance function
$$\varphi (x,y) =\sup \{ \varphi \mid ((1 - \varphi )x, (1 + \varphi )y) \in \mathcal{P}\},$$which is a special case of the general directional distance function proposed by Chambers et al. (1996). While this distance function can be estimated by linear programming methods, proofs of asymptotic properties such as consistency, rate of convergence, etc. remain elusive.
- 6.
In cases where CRS is assumed, the constraint \({\sum \nolimits }_{i=1}^{n}{\delta }_{i} = 1\) is omitted from (6.42). In such cases, efficiency is estimated in terms of distance to the boundary of the convex cone of the sample observations, as opposed to the convex hull (of the free-disposal hull) of the sample observations.
- 7.
Wheelock and Wilson (2008) gave a numerical algorithm for computing their unconditional, hyperbolic order-α quantile efficiency estimator. When α = 1, their estimator is equivalent to (6.61) and (6.62). Typically, computing \(\widehat{{H}}_{n}(x,y)\) using (6.61) will be faster than setting α = 1 and applying the numerical algorithm given in Wheelock and Wilson.
- 8.
- 9.
Recall that for any natural number d, a unit d-sphere is the set of points in (d + 1)-dimensional Euclidean space lying at distance one from a central point; the set of points comprises a d-dimensional manifold in Euclidean (d + 1)-space.
- 10.
The notation Exp(3) denotes an exponential distribution with parameter 3; hence \(E(\nu ) = 1/3\).
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Acknowledgements
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.
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Wilson, P.W. (2011). Asymptotic Properties of Some Non-Parametric Hyperbolic Efficiency Estimators. In: Van Keilegom, I., Wilson, P. (eds) Exploring Research Frontiers in Contemporary Statistics and Econometrics. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2349-3_6
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