Explaining Efficiency in Nonparametric Frontier Models: Recent Developments in Statistical Inference

  • Luiza Bădin
  • Cinzia Daraio


The explanation of efficiency differentials is an essential step in any frontier analysis study that aims to measure and compare the performance of decision making units. The conditional efficiency measures that have been introduced in recent years (Daraio and Simar, J. Prod. Anal. 24:93–121, 2005) represent an attractive alternative to two-step approaches, to handle external environmental factors, avoiding additional assumptions such as the separability between the input-output space and the space of external factors. Although affected by the curse of dimensionality, nonparametric estimation of conditional measures of efficiency eliminates any potential specification issue associated with parametric approaches. The nonparametric approach requires, however, estimation of a nonstandard conditional distribution function which involves smoothing procedures, and therefore the estimation of a bandwidth parameter. Recently, Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) proposed a data driven procedure for selecting the optimal bandwidth based on a general result obtained by Hall et al. (J. Am. Stat. Assoc. 99(486):1015–1026, 2004) for estimating conditional probability densities. The method employs least squares cross-validation (LSCV) to determine the optimal bandwidth with respect to a weighted integrated squared error (WISE) criterion.This paper revisits some of the recent advances in the literature on handling external factors in the nonparametric frontier framework. Following the Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) approach, we provide a detailed description of optimal bandwidth selection in nonparametric conditional efficiency estimation, when mixed continuous and discrete external factors are available. We further propose an heterogeneous bootstrap which allows improving the detection of the impact of the external factors on the production process, by computing pointwise confidence intervals on the ratios of conditional to unconditional efficiency measures.We illustrate these extensions through some simulated data and an empirical application using the sample of U.S. mutual funds previously analyzed in Daraio and Simar (J. Prod. Anal. 24:93–121, 2005; Eur. J. Oper. Res. 175(1):516–542, 2006; Advanced Robust and Nonparametric Methods in Efficiency Analysis: Methodology and Applications, Springer, New York, 2007a).


Mutual Fund Efficiency Score Nonparametric Estimator Optimal Bandwidth Conditional Measure 
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.



Previous discussions with Léopold Simar as well as his valuable comments and suggestions contributed a lot in improving the quality of the present paper and are gratefully acknowledged. The usual disclaimers apply.


  1. Aitchison, J., & Aitken, C.G.G. (1976). Multivariate binary discrimination by the kernel method. Biometrika, 63(3), 413–420.MathSciNetMATHCrossRefGoogle Scholar
  2. Banker, R.D., & Morey, R.C. (1986). Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34(4), 513–521.MATHCrossRefGoogle Scholar
  3. Bădin, L., Daraio, C., & Simar, L. (2010). Optimal bandwidth selection for conditional efficiency measures: a data-driven approach. European Journal of Operational Research, 201(2), 633–640.MATHCrossRefGoogle 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. Charnes, A., Cooper, W.W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444.MathSciNetMATHCrossRefGoogle Scholar
  6. Cooper, W.W., Seiford, L.M., & Tone, K. (2000). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software. Boston: Kluwer Academic Publishers.Google Scholar
  7. Daouia, A., & Simar, L. (2007). Nonparametric efficiency analysis: a multivariate conditional quantile approach. Journal of Econometrics, 140, 375–400.MathSciNetMATHCrossRefGoogle Scholar
  8. Daraio, C., & Simar, L. (2005). Introducing environmental variables in nonparametric frontier models: a probabilistic approach. The Journal of Productivity Analysis, 24, 93–121.CrossRefGoogle Scholar
  9. Daraio, C., & Simar, L. (2006). A robust nonparametric approach to evaluate and explain the performance of mutual funds. European Journal of Operational Research, 175(1), 516–542.MathSciNetMATHCrossRefGoogle Scholar
  10. Daraio, C., & Simar, L. (2007a). Advanced Robust and Nonparametric Methods in Efficiency Analysis. Methodology and applications. New York: Springer.MATHGoogle Scholar
  11. Daraio, C., & Simar, L. (2007b). Conditional nonparametric Frontier models for convex and non convex technologies: A unifying approach. Journal of Productivity Analysis, 28, 13–32.CrossRefGoogle Scholar
  12. Daraio, C., Simar, L., & Wilson, P. (2010). Testing whether two-stage estimation is meaningful in nonparametric models of production, Discussion Paper #1030, Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  13. Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor-efficiency in post offices. In M. Marchand, P. Pestieau, & H. Tulkens (eds.) The Performance of public enterprises – Concepts and Measurement (pp. 243–267). Amsterdam: North-Holland.Google Scholar
  14. Fan, J., & Gijbels, I. (1996). Local polinomial modelling and its applications. London: Chapman and Hall.Google Scholar
  15. Farrell, M.J. (1957). The measurement of the Productive Efficiency. Journal of the Royal Statistical Society, Series A, CXX, Part 3, 253–290.Google Scholar
  16. Färe, R., Grosskopf, S., & Lovell, C.A.K. (1994). Production frontiers. Cambridge: Cambridge University Press.Google Scholar
  17. 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
  18. Hall, P., Racine, J.S., Li, Q. (2004). Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association, 99(486), 1015–1026.MathSciNetMATHCrossRefGoogle Scholar
  19. Jeong, S.O., Park, B.U., & Simar, L. (2008). Nonparametric conditional efficiency measures: asymptotic properties. Annals of Operations Research, doi: 10.1007/s10479-008-0359-5.Google Scholar
  20. Li, Q., & Racine, J. (2007). Nonparametric econometrics: theory and practice. New Jersey: Princeton University Press.MATHGoogle Scholar
  21. Li, Q., & Racine, J. (2008). Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data. Journal of Business and Economic Statistics, 26(4), 423–434.MathSciNetCrossRefGoogle Scholar
  22. Li, Q., Racine, J., Wooldridge, J.M. (2009). Efficient estimation of average treatment effect with mixed categorical and continuous data. Journal of Business and Economic Statistics, 26(4), 423–434.MathSciNetMATHCrossRefGoogle Scholar
  23. Pagan, A., & Ullah, A. (1999). Nonparametric Econometrics. Cambridge: Cambridge University Press.Google Scholar
  24. Ouyang, D., Li, Q., & Racine, J. (2006). Cross-validation and the estimation of probability distributions with categorical data. Nonparametric Statistics, 18(1), 69–100.MathSciNetMATHCrossRefGoogle Scholar
  25. Park, B., Simar, L., & Weiner, C. (2000). The FDH estimator for productivity efficiency scores: asymptotic properties. Econometric Theory, 16, 855–877.MathSciNetMATHCrossRefGoogle Scholar
  26. Park, B., Simar, L., & Zelenyuk, V. (2008). Local likelihood estimation of truncated regression and its partial derivatives: Theory and application. Journal of Econometrics, 146(1), 185–198.MathSciNetCrossRefGoogle Scholar
  27. Simar, L., & Wilson, P.W. (2007). Estimation and inference in two-stage, semi-parametric models of production processes. Journal of Econometrics, 136(1), 31–64.MathSciNetCrossRefGoogle Scholar
  28. Simar, L., & Wilson, P.W. (2008). Statistical inference in nonparametric frontier models: recent developments and perspectives. In Harold O. Fried, C.A. Knox Lovell, & Shelton S. Schmidt (Eds.), The Measurement of Productive Efficiency, 2nd edn. Oxford: Oxford University Press.Google Scholar
  29. Simar, L., & Wilson, P.W. (2009). Inference by Subsampling in Nonparametric Frontier Models, Discussion Paper #0933, Institut de Statistique, Université Catholique de Louvain, Louvain- la-Neuve, Belgium.Google Scholar
  30. Simar, L., & Wilson, P.W. (2010). Two-Stage DEA: Caveat Emptor, Discussion Paper #10xx, Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.Google Scholar
  31. Titterington, D.M. (1980). A comparative study of kernel-based density estimates for categorical data. Technometrics, 22(2), 259–268.MathSciNetMATHCrossRefGoogle Scholar
  32. Wang, M.C., & Van Ryzin, J. (1981). A class of smooth estimators for discrete distributions. Biometrika, 68, 301–309.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Applied MathematicsBucharest Academy of Economic StudiesBucharestRomania
  2. 2.Department of Statistical InferenceGh. Mihoc - C. Iacob Institute of Mathematical Statistics and Applied MathematicsBucharestRomania
  3. 3.Department of Management, CIEG - Centro Studi di Ingegneria Economico-GestionaleUniversity of BolognaBolognaItaly

Personalised recommendations