nach oben

Erschienen in:

2012 | OriginalPaper | Buchkapitel

6. Asymptotic Properties of Some Non-Parametric Hyperbolic Efficiency Estimators

verfasst von : Paul W. Wilson

Erschienen in: Exploring Research Frontiers in Contemporary Statistics and Econometrics

Verlag: Physica-Verlag HD

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A hyperbolic measure of technical efficiency was proposed by Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 1985) wherein efficiency is measured by the simultaneous maximum, feasible reduction in input quantities and increase in output quantities. In cases where returns to scale are not constant, the non-parametric data envelopment analysis (DEA) estimator of hyperbolic efficiency cannot be written as a linear program; consequently, the measure has not been used in empirical studies except where returns to scale are constant, allowing the estimator to be computed by linear programming methods. This paper develops an alternative estimator of the hyperbolic measure proposed by Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 1985). Statistical consistency and rates of convergence are established for the new estimator. A numerical procedure allowing computation of the original estimator is provided, and this estimator is also shown to be consistent, with the same rate of convergence as the new estimator. In addition, an unconditional, hyperbolic order-m efficiency estimator is developed by extending the ideas of Cazals et al. (J. Econometric. 106:1–25, 2002). Asymptotic properties of this estimator are also given.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Panel Data, Factor Models, and the Solow Residual

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

See Simar and Wilson (2000b) for a survey, and Kneip et al. (2008) for more recent results, on the statistical properties of DEA estimators. See Simar and Wilson (1998, 2000a) and Kneip et al. (2008, 2011) for details about the use of bootstrap methods to make inferences based on DEA.

One can find numerous published applications of DEA to datasets with 50–150 observations and 5 or more dimensions in the input-output space. DEA-based inefficiency estimates from such studies are likely meaningless in a statistical sense due to the curse of dimensionality problem (see Simar and Wilson (2000b) for discussion).

Wheelock and Wilson (2008) also derived asymptotic properties for a hyperbolic FDH efficiency estimator.

The Farrell (1957) input and output distance functions defined in (6.7)–(6.9) are reciprocals of the corresponding Shephard (1970) measures.

Alternatively, one might consider estimating the directional distance function

$$\varphi (x,y) =\sup \{ \varphi \mid ((1 - \varphi )x, (1 + \varphi )y) \in \mathcal{P}\},$$

which is a special case of the general directional distance function proposed by Chambers et al. (1996). While this distance function can be estimated by linear programming methods, proofs of asymptotic properties such as consistency, rate of convergence, etc. remain elusive.

In cases where CRS is assumed, the constraint ${\sum \nolimits }_{i=1}^{n}{\delta }_{i} = 1$ is omitted from (6.42). In such cases, efficiency is estimated in terms of distance to the boundary of the convex cone of the sample observations, as opposed to the convex hull (of the free-disposal hull) of the sample observations.

Wheelock and Wilson (2008) gave a numerical algorithm for computing their unconditional, hyperbolic order-α quantile efficiency estimator. When α = 1, their estimator is equivalent to (6.61) and (6.62). Typically, computing $\widehat{{H}}_{n}(x,y)$ using (6.61) will be faster than setting α = 1 and applying the numerical algorithm given in Wheelock and Wilson.

To see this, replace x in (6.60) with w, set q = 1, y = 0, and replace p with (p + q). The resulting expression is equivalent to (6.65).

Recall that for any natural number d, a unit d-sphere is the set of points in (d + 1)-dimensional Euclidean space lying at distance one from a central point; the set of points comprises a d-dimensional manifold in Euclidean (d + 1)-space.

The notation Exp(3) denotes an exponential distribution with parameter 3; hence $E(\nu ) = 1/3$.

Aigner, D., Lovell, C.A.K., & Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6, 21–37.MathSciNetMATHCrossRef

Aragon, Y., Daouia, A., & Thomas-Agnan, C. (2005). Nonparametric frontier estimation: A conditional quantile-based approach. Econometric Theory, 21, 358–389.MathSciNetMATHCrossRef

Boos, D.D., & Serfling, R.J. (1980). A note on differentials and the CLT and LIL for statistical functions, with application to M-estimates. The Annals of Statistics, 8, 618–624.MathSciNetMATHCrossRef

Cazals, C., Florens, J.P., & Simar, L. (2002). Nonparametric frontier estimation: A robust approach. Journal of Econometrics, 106, 1–25.MathSciNetMATHCrossRef

Chambers, R.G., Chung, Y., & Färe, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70, 407–419.MATHCrossRef

Charnes, A., Cooper, W.W., & Rhodes, E. (1981). Evaluating program and managerial efficiency: An application of data envelopment analysis to program follow through. Management Science, 27, 668–697.CrossRef

Daouia, A. (2003). Nonparametric Analysis of Frontier Production Functions and Efficiency Measurement using Nonstandard Conditional Quantiles. PhD thesis, Groupe de Recherche en Economie Mathématique et Quantititative, Université des Sciences Sociales, Toulouse I, et Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse III, 2003.

Daouia, A., & Simar, L. (2007). Nonparametric efficiency analysis: A multivariate conditional quantile approach. Journal of Econometrics, 140, 375–400.MathSciNetMATHCrossRef

Daraio, C., & Simar, L. (2005). Introducing environmental variables in nonparametric frontier models: A probabilistic approach. Journal of Productivity Analysis, 24, 93–121.CrossRef

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. Amsterdam: North-Holland.

Färe, R. (1988). Fundamentals of Production Theory. Berlin: Springer.MATH

Färe, R., Grosskopf, S., & Lovell, C.A.K. (1985). The Measurement of Efficiency of Production. Boston: Kluwer-Nijhoff Publishing.

Farrell, M.J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society A, 120, 253–281.CrossRef

Gattoufi, S., Oral, M., & Reisman, A. (2004). Data envelopment analysis literature: A bibliography update (1951–2001). Socio-Economic Planning Sciences, 38, 159–229.CrossRef

Kneip, A., Park, B., & Simar, L. (1998). A note on the convergence of nonparametric DEA efficiency measures. Econometric Theory, 14, 783–793.MathSciNetCrossRef

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.MathSciNetMATHCrossRef

Kneip, A., Simar, L., & Wilson, P.W. (2011). A computationally efficient, consistent bootstrap for inference with non-parametric DEA estimators. Computational Economics, Institut de Statistique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.

Korostelev, A., Simar, L., & Tsybakov, A.B. (1995a). Efficient estimation of monotone boundaries. The Annals of Statistics, 23, 476–489.MathSciNetMATHCrossRef

Korostelev, A., Simar, L., & Tsybakov, A.B. (1995b). On estimation of monotone and convex boundaries. Publications de l’Institut de Statistique de l’Université de Paris XXXIX, 1, 3–18.MathSciNet

Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43, 645–646.MATHCrossRef

Muller, M.E. (1959). A note on a method for generating points uniformly on n-dimensional spheres. Communications of the Association for Computing Machinery, 2, 19–20.MATHCrossRef

Park, B.U., Simar, L., & Weiner, C. (2000). FDH efficiency scores from a stochastic point of view. Econometric Theory, 16, 855–877.MathSciNetMATHCrossRef

Shephard, R.W. (1970). Theory of Cost and Production Functions. Princeton: Princeton University Press.MATH

Simar, L., & Wilson, P.W. (1998). Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science, 44, 49–61.MATHCrossRef

Simar, L., & Wilson, P.W. (2000a). A general methodology for bootstrapping in non-parametric frontier models. Journal of Applied Statistics, 27, 779–802.MathSciNetMATHCrossRef

Simar, L., & Wilson, P.W. (2000b). Statistical inference in nonparametric frontier models: The state of the art. Journal of Productivity Analysis, 13, 49–78.CrossRef

Simar, L., & Wilson, P.W. (2011). Inference by the m out of n Bootstrap in Nonparametric Frontier Models. Journal of Productivity Analysis, 36, 33–53.CrossRef

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, 209–225.MathSciNetCrossRef

Wheelock, D.C., & Wilson, P.W. (2009). Robust nonparametric quantile estimation of efficiency and productivity change in U.S. commercial banking, 1985–2004. Journal of Business and Economic Statistics, 27, 354–368.

Titel: Asymptotic Properties of Some Non-Parametric Hyperbolic Efficiency Estimators
verfasst von: Paul W. Wilson
Verlag: Physica-Verlag HD
Buch: Exploring Research Frontiers in Contemporary Statistics and Econometrics
Print ISBN: 978-3-7908-2348-6

Electronic ISBN: 978-3-7908-2349-3

Copyright-Jahr: 2012
DOI: https://doi.org/10.1007/978-3-7908-2349-3_6

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"