Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-07T17:01:57.149Z Has data issue: false hasContentIssue false
  • English
  • Français

ROBUSTIFIED EXPECTED MAXIMUM PRODUCTION FRONTIERS

Published online by Cambridge University Press:  27 March 2020

Abdelaati Daouia ,
Jean-Pierre Florens  and
Léopold Simar
Abdelaati Daouia
Affiliation:
University of Toulouse Capitole
Jean-Pierre Florens
Affiliation:
University of Toulouse Capitole
Léopold Simar*
Affiliation:
Université Catholique de Louvain
*
Address correspondence to Léopold Simar, Institut de Statistique, Biostatistique et Sciences Actuarielles, Université Catholique de Louvain, Belgium; e-mail: leopold.simar@uclouvain.be.
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to construct a robust nonparametric estimator for the production frontier. We study this problem under a regression model with one-sided errors, where the regression function defines the achievable maximum output, for a given level of inputs-usage, and the regression error defines the inefficiency term. The main tool is a concept of partial regression boundary defined as a special probability-weighted moment. This concept motivates a robustified unconditional alternative to the pioneering class of nonparametric conditional expected maximum production functions. We prove that both the resulting benchmark partial frontier and its estimator share the desirable monotonicity of the true full frontier. We derive the asymptotic properties of the partial and full frontier estimators, and unravel their behavior from a robustness theory point of view. We provide numerical illustrations and Monte Carlo evidence that the presented concept of unconditional expected maximum production functions is more efficient and reliable in filtering out noise than the original conditional version. The methodology is very easy and fast to implement. Its usefulness is discussed through two concrete datasets from the sector of Delivery Services, where outliers are likely to affect the traditional conditional approach.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 2 , April 2021 , pp. 346 - 387
Copyright
© Cambridge University Press 2020

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.)

Footnotes

The authors would like to thank two referees and the Co-Editor for their valuable suggestions, which have significantly improved the paper. They acknowledge funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program, grant ANR-17-EURE-0010. This research was also supported by the French National Research Agency under the grant ANR-19-CE40-0013-01/ExtremReg project. Support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy) is also gratefully acknowledged.

References

REFERENCES

Aigner, D.J., Lovell, C.A.K. & Schmidt, P. 1977 Formulation and estimation of stochastic frontier models. Journal of Econometrics 6, 21–37.CrossRefGoogle Scholar
Aragon, Y., Daouia, A. & Thomas-Agnan, C. 2005 Nonparametric frontier estimation: A conditional quantile-based approach. Econometric Theory 21, 358–389.CrossRefGoogle Scholar
Cazals, C., Florens, J.P. & Simar, L. 2002 Nonparametric frontier estimation: A robust approach. Journal of Econometrics 106, 1–25.CrossRefGoogle Scholar
Daouia, A., Florens, J.P. & Simar, L. 2010 Frontier estimation and extreme value theory. Bernoulli 16(4), 1039–1063.CrossRefGoogle Scholar
Daouia, A., Florens, J.P. & Simar, L. 2012 Regularization of non-parametric frontier estimators. Journal of Econometrics 168, 285–299.CrossRefGoogle Scholar
Daouia, A., Gardes, L. & Girard, S. 2013 On kernel smoothing for extremal quantile regression. Bernoulli 19, 2557–2589.CrossRefGoogle Scholar
Daouia, A. & Gijbels, I. 2011 Robustness and inference in nonparametric partial frontier modeling. Journal of Econometrics 161, 147–165.CrossRefGoogle Scholar
Daouia, A., Girard, S. & Guillou, A. 2014 A Γ moment approach to monotonic boundary estimation. Journal of Econometrics 78, 727–740.CrossRefGoogle Scholar
Daouia, A., Noh, H. & Park, B.U. 2016 Data envelope fitting with constrained polynomial splines. Journal of the Royal Statistical Society: Series B 78(Part 1), 3–30.CrossRefGoogle Scholar
Daouia, A. & Ruiz-Gazen, A. 2006 Robust nonparametric frontier estimators: Qualitative robustness and influence function. Statistica Sinica 16, 1233–1253.Google Scholar
Daouia, A. & Simar, L. 2005 Robust nonparametric estimators of monotone boundaries. Journal of Multivariate Analysis 96, 311–331.CrossRefGoogle Scholar
Daouia, A. & Simar, L. 2007 Nonparametric efficiency analysis: A multivariate conditional quantile approach. Journal of Econometrics 140, 375–400.CrossRefGoogle Scholar
Daouia, A., Simar, L. & Wilson, P. 2017 Measuring firm performance by using nonparametric quantile-type distances. Econometric Reviews 36, 156–181.CrossRefGoogle Scholar
Deprins, D., Simar, L. & Tulkens, H. 1984 Measuring labor inefficiency in post offices. In Marchand, M., Pestieau, P. & Tulkens, H. (eds.) The Performance of Public Enterprises: Concepts and measurements, pp. 243–267. North-Holland.Google Scholar
Farrell, M.J. 1957 The measurement of productive efficiency. Journal of the Royal Statistical Society A 120, 253–281.CrossRefGoogle Scholar
Florens, J.P. & Simar, L. 2005 Parametric approximations of nonparametric frontier. Journal of Econometrics 124(1), 91–116.CrossRefGoogle Scholar
Gijbels, I., Mammen, E., Park, B.U. & Simar, L. 1999 On estimation of monotone and concave frontier functions. Journal of American Statistical Association 94, 220–228.Google Scholar
Gijbels, I. & Sznajder, D. 2013 Testing tail monotonicity by constrained copula estimation. Insurance: Mathematics and Economics 52(2), 338–351.Google Scholar
Greene, W.H. 2008 The econometric approach to efficiency analysis. In Fried, Harold, Lovell, C.A. Knox & Schmidt, Shelton (eds.), The Measurement of Productive Efficiency, 2nd edn. Oxford University Press.Google Scholar
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J. & Stahel, W.A. 1986 Robust Statistics: The Approach Based on Influence Functions. Wiley.Google Scholar
Kneip, A., Simar, L. & Van Keilegom, I. 2015 Frontier estimation in the presence of measurement error with unknown variance. Journal of Econometrics 184, 379–393.CrossRefGoogle Scholar
Kneip, A, Simar, L. & Wilson, P.W. 2008 Asymptotics and consistent bootstraps for DEA estimators in non-parametric frontier models. Econometric Theory 24, 1663–1697.CrossRefGoogle Scholar
Kumbhakar, S.C., Park, B.U., Simar, L. & Tsionas, E.G. 2007 Nonparametric stochastic frontiers: A local likelihood approach. Journal of Econometrics 137(1), 1–27.CrossRefGoogle Scholar
Meeusen, W. & van den Broek, J. 1977 Efficiency estimation from Cobb-Douglas production function with composed error. International Economic Review 8, 435–444.CrossRefGoogle Scholar
Noh, H. 2014 Frontier estimation using kernel smoothing estimators with data transformation. Journal of the Korean Statistical Society 43, 503–512.CrossRefGoogle Scholar
Park, B. Simar, L. & Weiner, C. 2000 The FDH estimator for productivity efficiency scores: Asymptotic properties. Econometric Theory 16, 855–877.CrossRefGoogle Scholar
Rousseeuw, P.J. 1981 A new infinitesimal approach to robust estimation. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 56, 127–132.CrossRefGoogle Scholar
Serfling, R.T. 1980 Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Simar, L. 2003 Detecting outliers in frontier models: A simple approach. Journal of Productivity Analysis 20, 391–424.CrossRefGoogle Scholar
Simar, L. & Zelenyuk, V. 2011 Stochastic FDH/DEA estimators for frontier analysis. Journal of Productivity Analysis 36, 1–20.CrossRefGoogle Scholar
van der Vaart, A.W. 1998 Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics, 3. Cambridge University Press.Google Scholar
van der Vaart, A.W. & Wellner, J.A. 1996 Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics. Springer-Verlag.Google Scholar
Wheelock, D.C. & Wilson, P.W. 2008 Non-parametric, unconditional quantile estimation for efficiency analysis with an application to federal reserve check processing operations. Journal of Econometrics 145(1–2), 209–225.CrossRefGoogle Scholar