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On estimation of monotone and concave frontier functions

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  • GIJBELS, Irène
  • MAMMEN, Enno
  • PARK, Byeong U.
  • SIMAR, Léopold

Abstract

A way for measuring the efficiency of enterprises is via the estimation of the so-called production frontier, which is the upper boundary of the support of the population density in the input and output space. It is reasonable to assume that the production frontier is a concave monotone function. Then, a famous estimator is the data envelopment analysis (DEA) estimator, which is the lowest concave monotone increasing function covering all sample points. This estimator is biased downwards since it never exceeds the true production frontier. In this paper we derive the asymptotic distribution of the DEA estimator, which enables us to assess the asymptotic bias and hence to propose an improved bias corrected estimator. This bias corrected estimator involves consistent estimation of the density function as well as of the second derivative of the production frontier. We also discuss briefly the construction of asymptotic confidence intervals. The finite sample performance of the bias corrected estimator is investigated via a simulation study and the procedure is illustrated for a real data example.
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Suggested Citation

  • GIJBELS, Irène & MAMMEN, Enno & PARK, Byeong U. & SIMAR, Léopold, 1999. "On estimation of monotone and concave frontier functions," CORE Discussion Papers RP 1392, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvrp:1392
    Note: In : Journal of the American Statistical Association, 94(445), 220-228, 1999
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    File URL: http://dx.doi.org/10.1080/01621459.1999.10473837
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    References listed on IDEAS

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    1. Greene, William H., 1980. "Maximum likelihood estimation of econometric frontier functions," Journal of Econometrics, Elsevier, vol. 13(1), pages 27-56, May.
    2. Tsybakov, A.B. & Korostelev, A.P. & Simar, L., 1992. "Efficient Estimation of Monotone Boundaries," Papers 9209, Catholique de Louvain - Institut de statistique.
    3. Greene, William H., 1990. "A Gamma-distributed stochastic frontier model," Journal of Econometrics, Elsevier, vol. 46(1-2), pages 141-163.
    4. Hardle, W. & Park, B. U. & Tsybakov, A. B., 1995. "Estimation of Non-sharp Support Boundaries," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 205-218, November.
    5. Berndt, Ernst R. & Christensen, Laurits R., 1973. "The translog function and the substitution of equipment, structures, and labor in U.S. manufacturing 1929-68," Journal of Econometrics, Elsevier, vol. 1(1), pages 81-113, March.
    6. Christensen, Laurits R & Greene, William H, 1976. "Economies of Scale in U.S. Electric Power Generation," Journal of Political Economy, University of Chicago Press, vol. 84(4), pages 655-676, August.
    7. Rajiv D. Banker, 1993. "Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation," Management Science, INFORMS, vol. 39(10), pages 1265-1273, October.
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