Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T18:50:32.653Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTICS AND CONSISTENT BOOTSTRAPS FOR DEA ESTIMATORS IN NONPARAMETRIC FRONTIER MODELS

Published online by Cambridge University Press:  17 July 2008

Alois Kneip ,
Léopold Simar  and
Paul W. Wilson
Alois Kneip
Affiliation:
Universität Bonn
Léopold Simar
Affiliation:
Université Catholique de Louvain
Paul W. Wilson*
Affiliation:
Clemson University
*
Address correspondence to Paul W. Wilson, Department of Economics, 222 Sirrine Hall, Clemson University, Clemson, South Carolina 29634, USA; e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonparametric data envelopment analysis (DEA) estimators based on linear programming methods have been widely applied in analyses of productive efficiency. The distributions of these estimators remain unknown except in the simple case of one input and one output, and previous bootstrap methods proposed for inference have not been proved consistent, making inference doubtful. This paper derives the asymptotic distribution of DEA estimators under variable returns to scale. This result is used to prove consistency of two different bootstrap procedures (one based on subsampling, the other based on smoothing). The smooth bootstrap requires smoothing the irregularly bounded density of inputs and outputs and smoothing the DEA frontier estimate. Both bootstrap procedures allow for dependence of the inefficiency process on output levels and the mix of inputs in the case of input-oriented measures, or on input levels and the mix of outputs in the case of output-oriented measures.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 6 , December 2008 , pp. 1663 - 1697
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Banker, R.D. (1993) Maximum likelihood, consistency and data envelopment analysis: A statistical foundation. Management Science 39, 1265–1273.CrossRefGoogle Scholar
Bickel, P.J. & Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. Annals of Statistics 9, 1196–1217.CrossRefGoogle Scholar
Charnes, A., Cooper, W.W., & Rhodes, E. (1978) Measuring the efficiency of decision making units. European Journal of Operational Research 2, 429–444.CrossRefGoogle Scholar
Charnes, A., Cooper, W.W., & Rhodes, E. (1979) Measuring the efficiency of decision making units. European Journal of Operational Research 3, 339.CrossRefGoogle Scholar
Debreu, G. (1951) The coefficient of resource utilization. Econometrica 19, 273–292.CrossRefGoogle Scholar
Färe, R. (1988) Fundamentals of Production Theory. Springer-Verlag.CrossRefGoogle Scholar
Färe, R., Grosskopf, S., & Lovell, C.A.K. (1985) The Measurement of Efficiency of Production. Kluwer-Nijhoff.CrossRefGoogle Scholar
Farrell, M.J. (1957) The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A 120, 253–281.CrossRefGoogle Scholar
Gijbels, I., Mammen, E., Park, B.U., & Simar, L. (1999) On estimation of monotone and concave frontier functions. Journal of the American Statistical Association 94, 220–228.CrossRefGoogle Scholar
Hall, P. (1992) The Bootstrap and Edgeworth Expansion. Springer-Verlag.CrossRefGoogle Scholar
Kneip, A., Park, B., & Simar, L. (1998) A note on the convergence of nonparametric DEA efficiency measures. Econometric Theory 14, 783–793.CrossRefGoogle Scholar
Korostelev, A., Simar, L., & Tsybakov, A.B. (1995) On estimation of monotone and convex boundaries. Publications de l'Institut de Statistique de l'Université de Paris XXXIX 1, 3–18.Google Scholar
Sheather, S.J. & Jones, M.C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B 53, 683–690.Google Scholar
Shephard, R.W. (1970) Theory of Cost and Production Functions. Princeton University Press.Google Scholar
Simar, L. (1996) Aspects of statistical analysis in DEA-type frontier models. Journal of Productivity Analysis 7, 177–185.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (1998) Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science 44, 49–61.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (1999a) Of course we can bootstrap DEA scores! but does it mean anything? Logic trumps wishful thinking. Journal of Productivity Analysis 11, 93–97.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (1999b) Some problems with the Ferrier/Hirschberg bootstrap idea. Journal of Productivity Analysis 11, 67–80.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (2000a) A general methodology for bootstrapping in non-parametric frontier models. Journal of Applied Statistics 27, 779–802.CrossRefGoogle Scholar
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
Swanepoel, J.W.H. (1986) A note on proving that the (modified) bootstrap works. Communications in Statistics: Theory and Methods 15, 3193–3203.CrossRefGoogle Scholar