Nonparametric Frontier Estimation from Noisy Data

  • Maik Schwarz
  • Sébastien Van Bellegem
  • Jean-Pierre Florens


A new nonparametric estimator of production frontiers is defined and studied when the data set of production units is contaminated by measurement error. The measurement error is assumed to be an additive normal random variable on the input variable, but its variance is unknown. The estimator is a modification of the m-frontier, which necessitates the computation of a consistent estimator of the conditional survival function of the input variable given the output variable. In this paper, the identification and the consistency of a new estimator of the survival function is proved in the presence of additive noise with unknown variance. The performance of the estimator is also studied using simulated data.


Data Envelopment Analysis Survival Function Production Unit Consistent Estimator Production Frontier 
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 supported by the “Agence National de la Recherche” under contract ANR-09-JCJC-0124-01 and by the IAP research network nr P6/03 of the Belgian Government (Belgian Science Policy). Comments from Ingrid Van Keilegom and an anonymous referee were most helpful to improve the final version of the manuscript. The usual disclaimer applies.


  1. Bigot, J., & Van Bellegem, S. (2009). Log-density deconvolution by wavelet thresholding. Scandinavian Journal of Statistics, 36, 749–763.MATHCrossRefGoogle Scholar
  2. Butucea, C., & Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric deconvolution model. Bernoulli, 11, 309–340.MathSciNetMATHCrossRefGoogle Scholar
  3. Butucea, C., Matias, C., & Pouet, C. (2008). Adaptivity in convolution models with partially known noise distribution. Electronic Journal of Statistics, 2, 897–915.MathSciNetCrossRefGoogle Scholar
  4. Carroll, R., & Hall, P. (1988). Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association, 83, 1184–1186.MathSciNetMATHCrossRefGoogle Scholar
  5. Cazals, C., Florens, J. P., & Simar, L. (2002). Nonparametric frontier estimation: a robust approach. Journal of Econometrics, 106, 1–25.MathSciNetMATHCrossRefGoogle Scholar
  6. Daouia, A., Florens, J., & Simar, L. (2009). Regularization in nonparametric frontier estimators (Discussion Paper No. 0922). Université catholique de Louvain, Belgium: Institut de statistique, biostatistique et sciences actuarielles.Google Scholar
  7. Daskovska, A., Simar, L., & Van Bellegem, S. (2010). Forecasting the Malmquist productivity index. Journal of Productivity Analysis, 33, 97–107.CrossRefGoogle Scholar
  8. De Borger, B., Kerstens, K., Moesen, W., & Vanneste, J. (1994). A non-parametric free disposal hull (FDH) approach to technical efficiency: an illustration of radial and graph efficiency measures and some sensitivity results. Swiss Journal of Economics and Statistics, 130, 647–667.Google Scholar
  9. Delaigle, A., Hall, P., & Meister, A. (2008). On deconvolution with repeated measurements. The Annals of Statistics, 36, 665–685.MathSciNetMATHCrossRefGoogle Scholar
  10. Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor inefficiency in post offices. In M. Marchand, P. Pestieau, & H. Tulkens (Eds.), The performance of public enterprises: Concepts and measurements (pp. 243–267). Amsterdam: North-Holland.Google Scholar
  11. Fan, J. (1991). On the optimal rate of convergence for nonparametric deconvolution problems. The Annals of Statistics, 19, 1257–1272.MathSciNetMATHCrossRefGoogle Scholar
  12. Färe, R., Grosskopf, S., & Knox Lovell, C. (1985). The measurements of efficiency of production (Vol. 6). New York: Springer.Google Scholar
  13. Hall, P., & Simar, L. (2002). Estimating a changepoint, boundary, or frontier in the presence of observation error. Journal of the American Statistical Association, 97, 523-534.MathSciNetMATHCrossRefGoogle Scholar
  14. Johannes, J., & Schwarz, M. (2009). Adaptive circular deconvolution by model selection under unknown error distribution (Discussion Paper No. 0931). Université catholique de Louvain, Belgium: Institut de statistique, biostatistique et sciences actuarielles.Google Scholar
  15. Johannes, J., Van Bellegem, S., & Vanhems, A. (2010). Convergence rates for ill-posed inverse problems with an unknown operator. Econometric Theory. (forthcoming)Google Scholar
  16. Johnstone, I., Kerkyacharian, G., Picard, D., & Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. Journal of the Royal Statistical Society Series B, 66, 547–573.MathSciNetMATHCrossRefGoogle Scholar
  17. Kneip, A., Park, B., & Simar, L. (1998). A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory, 14, 783–793.MathSciNetCrossRefGoogle Scholar
  18. Leleu, H. (2006). A linear programming framework for free disposal hull technologies and cost functions: Primal and dual models. European Journal of Operational Research, 168, 340–344.MathSciNetMATHCrossRefGoogle Scholar
  19. Li, T., & Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis, 65, 139–165.MathSciNetMATHCrossRefGoogle Scholar
  20. Meister, A. (2006). Density estimation with normal measurement error with unknown variance. Statistica Sinica, 16, 195–211.MathSciNetMATHGoogle Scholar
  21. Meister, A. (2007). Deconvolving compactly supported densities. Mathematical Methods in Statistics, 16, 195–211.Google Scholar
  22. Meister, A., Stadtmüller, U., & Wagner, C. (2010). Density deconvolution in a two-level heteroscedastic model with unknown error density. Electronic Journal of Statistics, 4, 36–57.MathSciNetCrossRefGoogle Scholar
  23. Neumann, M. H. (2007). Deconvolution from panel data with unknown error distribution. Journal of Multivariate Analysis, 98, 1955–1968.MathSciNetMATHCrossRefGoogle Scholar
  24. Park, B. U., Sickles, R. C., & Simar, L. (2003). Semiparametric efficient estimation of AR(1) panel data models. Journal of Econometrics, 117, 279-311.MathSciNetMATHCrossRefGoogle Scholar
  25. Park, B. U., Simar, L., & Weiner, C. (2000). The FDH estimator for productivity efficiency scores: asymptotic properties. Econometric Theory, 16, 855–877.MathSciNetMATHCrossRefGoogle Scholar
  26. Pensky, M., & Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. The Annals of Statistics, 27, 2033–2053.MathSciNetMATHCrossRefGoogle Scholar
  27. Schwarz, M., & Van Bellegem, S. (2010). Consistent density deconvolution under partially known error distribution. Statistics and Probability Letters, 80, 236–241.MathSciNetMATHCrossRefGoogle Scholar
  28. Seiford, L., & Thrall, R. (1990). Recent developments in DEA: The mathematical programming approach to frontier analysis. Journal of Econometrics, 46, 7–38.MathSciNetMATHCrossRefGoogle Scholar
  29. Shephard, R. W. (1970). Theory of cost and production functions. Princeton, NJ: Princeton University Press.MATHGoogle Scholar
  30. Simar, L. (2007). How to improve the performances of DEA/FDH estimators in the presence of noise? Journal of Productivity Analysis, 28, 183–201.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Maik Schwarz
    • 1
  • Sébastien Van Bellegem
    • 2
    • 3
  • Jean-Pierre Florens
    • 2
  1. 1.Institut de statistique, biostatistique et sciences actuariellesUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Toulouse School of Economics (GREMAQ)University of ToulouseToulouseFrance
  3. 3.Center for Operations Research and EconometricsUniversité catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations