Asymptotic Properties of Some Non-Parametric Hyperbolic Efficiency Estimators

  • Paul W. Wilson


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.


Data Envelopment Analysis Technical Efficiency Production Frontier Directional Distance Function Data Envelopment Analysis Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. Aigner, D., Lovell, C.A.K., & Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6, 21–37.MathSciNetMATHCrossRefGoogle Scholar
  2. Aragon, Y., Daouia, A., & Thomas-Agnan, C. (2005). Nonparametric frontier estimation: A conditional quantile-based approach. Econometric Theory, 21, 358–389.MathSciNetMATHCrossRefGoogle Scholar
  3. Boos, D.D., & Serfling, R.J. (1980). A note on differentials and the CLT and LIL for statistical functions, with application to M-estimates. The Annals of Statistics, 8, 618–624.MathSciNetMATHCrossRefGoogle Scholar
  4. Cazals, C., Florens, J.P., & Simar, L. (2002). Nonparametric frontier estimation: A robust approach. Journal of Econometrics, 106, 1–25.MathSciNetMATHCrossRefGoogle Scholar
  5. Chambers, R.G., Chung, Y., & Färe, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70, 407–419.MATHCrossRefGoogle Scholar
  6. Charnes, A., Cooper, W.W., & Rhodes, E. (1981). Evaluating program and managerial efficiency: An application of data envelopment analysis to program follow through. Management Science, 27, 668–697.CrossRefGoogle Scholar
  7. Daouia, A. (2003). Nonparametric Analysis of Frontier Production Functions and Efficiency Measurement using Nonstandard Conditional Quantiles. PhD thesis, Groupe de Recherche en Economie Mathématique et Quantititative, Université des Sciences Sociales, Toulouse I, et Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse III, 2003.Google Scholar
  8. Daouia, A., & Simar, L. (2007). Nonparametric efficiency analysis: A multivariate conditional quantile approach. Journal of Econometrics, 140, 375–400.MathSciNetMATHCrossRefGoogle Scholar
  9. Daraio, C., & Simar, L. (2005). Introducing environmental variables in nonparametric frontier models: A probabilistic approach. Journal of Productivity Analysis, 24, 93–121.CrossRefGoogle Scholar
  10. Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor inefficiency in post offices. In Marchand, M., Pestieau, P., &Tulkens, H. (Eds.). The Performance of Public Enterprises: Concepts and Measurements, pp. 243–267. Amsterdam: North-Holland.Google Scholar
  11. Färe, R. (1988). Fundamentals of Production Theory. Berlin: Springer.MATHGoogle Scholar
  12. Färe, R., Grosskopf, S., & Lovell, C.A.K. (1985). The Measurement of Efficiency of Production. Boston: Kluwer-Nijhoff Publishing.Google Scholar
  13. Farrell, M.J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society A, 120, 253–281.CrossRefGoogle Scholar
  14. Gattoufi, S., Oral, M., & Reisman, A. (2004). Data envelopment analysis literature: A bibliography update (1951–2001). Socio-Economic Planning Sciences, 38, 159–229.CrossRefGoogle Scholar
  15. Kneip, A., Park, B., & Simar, L. (1998). A note on the convergence of nonparametric DEA efficiency measures. Econometric Theory, 14, 783–793.MathSciNetCrossRefGoogle Scholar
  16. Kneip, A., Simar, L., & Wilson, P.W. (2008). Asymptotics and consistent bootstraps for DEA estimators in non-parametric frontier models. Econometric Theory, 24, 1663–1697.MathSciNetMATHCrossRefGoogle Scholar
  17. Kneip, A., Simar, L., & Wilson, P.W. (2011). A computationally efficient, consistent bootstrap for inference with non-parametric DEA estimators. Computational Economics, Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  18. Korostelev, A., Simar, L., & Tsybakov, A.B. (1995a). Efficient estimation of monotone boundaries. The Annals of Statistics, 23, 476–489.MathSciNetMATHCrossRefGoogle Scholar
  19. Korostelev, A., Simar, L., & Tsybakov, A.B. (1995b). On estimation of monotone and convex boundaries. Publications de l’Institut de Statistique de l’Université de Paris XXXIX, 1, 3–18.MathSciNetGoogle Scholar
  20. Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43, 645–646.MATHCrossRefGoogle Scholar
  21. Muller, M.E. (1959). A note on a method for generating points uniformly on n-dimensional spheres. Communications of the Association for Computing Machinery, 2, 19–20.MATHCrossRefGoogle Scholar
  22. Park, B.U., Simar, L., & Weiner, C. (2000). FDH efficiency scores from a stochastic point of view. Econometric Theory, 16, 855–877.MathSciNetMATHCrossRefGoogle Scholar
  23. Shephard, R.W. (1970). Theory of Cost and Production Functions. Princeton: Princeton University Press.MATHGoogle Scholar
  24. Simar, L., & Wilson, P.W. (1998). Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science, 44, 49–61.MATHCrossRefGoogle Scholar
  25. Simar, L., & Wilson, P.W. (2000a). A general methodology for bootstrapping in non-parametric frontier models. Journal of Applied Statistics, 27, 779–802.MathSciNetMATHCrossRefGoogle Scholar
  26. Simar, L., & Wilson, P.W. (2000b). Statistical inference in nonparametric frontier models: The state of the art. Journal of Productivity Analysis, 13, 49–78.CrossRefGoogle Scholar
  27. Simar, L., & Wilson, P.W. (2011). Inference by the m out of n Bootstrap in Nonparametric Frontier Models. Journal of Productivity Analysis, 36, 33–53.CrossRefGoogle Scholar
  28. Wheelock, D.C., & Wilson, P.W. (2008). Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations. Journal of Econometrics, 145, 209–225.MathSciNetCrossRefGoogle Scholar
  29. Wheelock, D.C., & Wilson, P.W. (2009). Robust nonparametric quantile estimation of efficiency and productivity change in U.S. commercial banking, 1985–2004. Journal of Business and Economic Statistics, 27, 354–368.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of EconomicsClemson UniversityClemsonUSA

Personalised recommendations