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Journal of Productivity Analysis

, Volume 13, Issue 1, pp 49–78 | Cite as

Statistical Inference in Nonparametric Frontier Models: The State of the Art

  • Léopold Simar
  • Paul W. Wilson
Article

Abstract

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.

DEA FDH Nonparametric estimation Efficiency Frontier models Bootstrapping 

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References

  1. Beran, R., and G. Ducharme. (1991). Asymptotic Theory for Bootstrap Methods in Statistics. Montreal: Centre de Reserches Mathematiques, University of Montreal.Google Scholar
  2. Banker, R. D. (1993). "Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation." Management Science 39(10), 1265-1273.Google Scholar
  3. Charnes, A., W. W. Cooper, and E. Rhodes. (1978). "Measuring the Inefficiency of Decision Making Units." European Journal of Operational Research 2(6), 429-444.Google Scholar
  4. Debreu, G. (1951). "The Coefficient of Resource Utilization." Econometrica 19(3), 273-292.Google Scholar
  5. Deprins, D., L. Simar, and H. Tulkens. (1984). "Measuring Labor Inefficiency in Post Offices." In M. Marchand, P. Pestieau, and H. Tulkens (eds.), The Performance of Public Enterprises: Concepts and Measurements. Amsterdam: North-Holland, 243-267.Google Scholar
  6. Efron, B. (1979). "Bootstrap Methods: Another Look at the Jackknife." Annals of Statistics 7, 1-16.Google Scholar
  7. Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, #38. Philadelphia: SIAM.Google Scholar
  8. Efron, B., and R. J. Tibshirani. (1993). An Introduction to the Bootstrap. London: Chapman and Hall.Google Scholar
  9. Farrell, M. J. (1957). "The Measurement of Productive Efficiency." Journal of the Royal Statistical Society A 120, 253-281.Google Scholar
  10. Ferrier, G. D., and J. G. Hirschberg. (1997). "Bootstrapping Confidence Intervals for Linear Programming Efficiency Scores: With an Illustration Using Italian Bank Data." Journal of Productivity Analysis 8, 19-33.Google Scholar
  11. Gijbels, I., E. Mammen, B. U. Park, and L. Simar. (1999). "On Estimation of Monotone and Concave Frontier Functions." Journal of the American Statistical Association 94, 220-228.Google Scholar
  12. Grosskopf, S. (1996). "Statistical Inference and Nonparametric Efficiency: A Selective Survey." Journal of Productivity Analysis 7, 161-176.Google Scholar
  13. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.Google Scholar
  14. Kneip, A., and L. Simar. (1996). "A General Framework for Frontier Estimation with Panel Data." Journal of Productivity Analysis 7, 187-212.Google Scholar
  15. Kneip, A., B. U. Park, and L. Simar. (1998). "A Note on the Convergence of Nonparametric DEA Estimators for Production Efficiency Scores." Econometric Theory 14, 783-793.Google Scholar
  16. Koopmans, T. C. (1951). "An Analysis of Production as an Efficient Combination of Activities." In T. C. Koopmans (ed.), Activity Analysis of Production and Allocation. Cowles Commission for Research in Economics, Monograph 13. New York: John-Wiley and Sons, Inc.Google Scholar
  17. Korostelev, A., L. Simar, and A. B. Tsybakov. (1995a). "Efficient Estimation of Monotone Boundaries." The Annals of Statistics 23, 476-489.Google Scholar
  18. Korostelev, A., L. Simar, and A. B. Tsybakov. (1995b). "On Estimation of Monotone and Convex Boundaries." Publications de l'Institut de Statistique des Universit´es de Paris XXXIX 1, 3-18.Google Scholar
  19. Lewis, P. A., A. S. Goodman, and J. M. Miller. (1969). "APseudo-Random Number Generator for the System/360." IBM Systems Journal 8, 136-146.Google Scholar
  20. Löthgren, M. (1997). "Bootstrapping the Malmquist Productivity Index: A Simulation Study." Working paper series in Economics and Finance #204, Department of Economic Statistics, Stockholm School of Economics, Sweden: forthcoming in Applied Economics Letters.Google Scholar
  21. Löthgren, M. (1998). "How to Bootstrap DEA Estimators: A Monte Carlo Comparison." (contributed paper presented at the Third Biennial Georgia Productivity Workshop, University of Georgia, Athens, GA, October 1998), Working paper series in Economics and Finance #223, Department of Economic Statistics, Stockholm School of Economics, Sweden.Google Scholar
  22. Löthgren, M., and M. Tambour. (1996). "Scale Efficiency and Scale Elasticity in DEA Models-A Bootstrapping Approach." Working paper series in Economics and Finance #91, Department of Economic Statistics, Stockholm School of Economics, Sweden: forthcoming in Applied Economics.Google Scholar
  23. Löthgren, M., and M. Tambour. (1997). "Bootstrapping the DEA-based Malmquist Productivity Index." in Magnus Tambour (Ph.D. dissertation), Essays on Performance Measurement in Health Care. Stockholm School of Economics, Stockholm, Sweden: forthcoming in Applied Economics.Google Scholar
  24. Manski, C. F. (1988). Analog Estimation Methods in Econometrics. New York: Chapman and Hall.Google Scholar
  25. Park, B., L. Simar, and C. Weiner. (1999). "The FDH Estimator for Productivity Efficiency Scores: Asymptotic Properties." Econometric Theory, forthcoming.Google Scholar
  26. Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. (1986). Numerical Recipes. Cambridge: Cambridge University Press.Google Scholar
  27. Shephard, R. W. (1970). Theory of Cost and Production Function. Princeton: Princeton University Press.Google Scholar
  28. Simar, L. (1992). "Estimating Efficiencies from Frontier Models with Panel Data: A Comparison of Parametric, Non-Parametric and Semi-Parametric Methods with Bootstrapping." Journal of Productivity Analysis 3, 167- 203.Google Scholar
  29. Simar, L. (1996). "Aspects of Statistical Analysis in DEA-Type Frontier Models." Journal of Productivity Analysis 7, 177-185.Google Scholar
  30. Simar, L., and P. W. Wilson. (1998a). "Sensitivity Analysis of Efficiency Scores: How to Bootstrap in Nonparametric Frontier Models." Management Science 44(11), 49-61.Google Scholar
  31. Simar, L., and P.W. Wilson. (1998b). "Nonparametric Tests of Returns to Scale, Discussion paper #9814." Institut de Statistique and CORE, Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  32. Simar, L., and P. W. Wilson. (2000). "A General Methodology for Bootstrapping in Nonparametric Frontier Models." Journal of Applied Statistics, forthcoming.Google Scholar
  33. Simar, L., and P. W. Wilson. (1999a). "Some Problems with the Ferrier=Hirschberg Bootstrap Idea." Journal of Productivity Analysis 11, 67-80.Google Scholar
  34. Simar, L., and P. W. Wilson. (1999b). "Of Course We Can Bootstrap DEA Scores! But Does It Mean Anything? Logic Trumps Wishful Thinking." Journal of Productivity Analysis 11, 93-97.Google Scholar
  35. Simar, L., and P. W. Wilson. (1999c). "Estimating and Bootstrapping Malmquist Indices." European Journal of Operations Research 115, 459-471.Google Scholar
  36. Weiner, H. C.M. (1998). Nonparametric Statistical Analysis of Productivity and Efficiency with the Free Disposal Hull. Doctoral Thesis, Institut de Statistique, Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  37. Wheelock, D. C., and P. W. Wilson. (1995). "Explaining Bank Failures: Deposit Insurance, Regulation, and Efficiency." Review of Economics and Statistics 77, 689-700.Google Scholar
  38. Wheelock, D. C., and P.W. Wilson. (2000). "Why Do Banks Disappear? The Determinants of US Bank Failures and Acquisitions." Review of Economics and Statistics, forthcoming.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Léopold Simar
    • 1
  • Paul W. Wilson
    • 2
  1. 1.Institut de StatistiqueUniversité Catholique de LouvainLouvain-La-NeuveBelgium
  2. 2.Department of EconomicsUniversity of TexasAustinUSA

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