Abstract
This chapter further examines the bootstrap method proposed by Simar and Wilson (Manag Sci 44(11):49–61, 1998) for DEA efficiency estimators. Some simplifications as well as Monte Carlo evidence on the coverage probabilities of confidence intervals estimated by the method are offered. In addition, we present similar evidence for confidence intervals estimated with the so-called naive bootstrap to illustrate the fact that the naive bootstrap is inconsistent in the DEA setting. Finally, we propose an iterated version of the bootstrap which may be used to improve bootstrap estimates of confidence intervals.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Throughout, inequalities involving vectors are defined on an element-by-element basis; e.g., for, \( \tilde{x} \), x \( \in \) \( R_{ + }^p \) means that some, but perhaps not all or none, of the corresponding elements of \( \tilde{x} \) and x may be equal, while some (but perhaps not all or none) of the elements of \( \tilde{x} \) may be greater than corresponding elements of x.
- 2.
Our characterization of the smoothness condition here is stronger than required; Kneip et al. (1998) require only Lipschitz continuity for \( D(x,y\left| {{ }P)} \right. \), which is implied by the simpler, but stronger requirement presented here.
- 3.
Banker (1993) showed, for the case q = 1, p ≥ 1, that \( \hat{P} \) is a consistent estimator of P, but did not provide convergence rates.
- 4.
In particular, the unknown quantities are determined by the curvature of \( {P^{\,\partial }} \) and the value of f(x, y) at the point where (x, y) is projected onto \( {P^{\,\partial }} \) in the direction orthogonal to x. See Gijbels et al. (1999) for additional details.
- 5.
The mean-square error of the bias-corrected estimator in (10.33) could be evaluated in a second-level bootstrap along the same lines as the iterated bootstrap we propose below in Sect. 10.9. See Efron and Tibshirani (1993, pp. 138) for a simple example in a different context.
- 6.
Explicit descriptions of why either variation of the naive bootstrap results in inconsistent estimates are given in Simar and Wilson (1999a, 2000a). Löthgren and Tambour (1997, 1999) and Löthgren (1998, 1999) employ a bizarre, illogical variant of the naive bootstrap different from the more typical variations we have mentioned. This approach also leads to an inconsistency problem, as discussed and confirmed with Monte Carlo experiments in Simar and Wilson (2000a).
- 7.
Appropriately chosen high-order kernels can reduce the order of the bias in the kernel density estimator, but run the risk of producing negative density estimates at some locations.
- 8.
This is the sense in which the kernel density estimator is a smoother, since it is, in effect, smoothing the empirical density function which places probability mass 1/n at each observed datum. Setting h = 0 in (10.34) yields the empirical density function, while letting h \( \to \infty \) yields a flat density estimate. The requirement that \( h = O({n^{{ - 1/5}}}) \) to ensure consistency of the kernel density estimate results from the fact that as n increases, h must become smaller, but not too quickly.
References
Banker RD. Maximum likelihood, consistency and data envelopment analysis: a statistical foundation. Manag Sci. 1993;39(10):1265–73.
Banker RD, Charnes A, Cooper WW. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci. 1984;30:1078–92.
Beran R, Ducharme G. Asymptotic theory for bootstrap methods in statistics. Montreal, QC: Centre de Reserches Mathematiques, University of Montreal; 1991.
Bickel PJ, Freedman DA. Some asymptotic theory for the bootstrap. Ann Stat. 1981;9:1196–217.
Chambers RG, Chung Y, Färe R. Benefit and distance functions. Econ Theor. 1996;70:407–19.
Charnes A, Cooper WW, Rhodes E. Measuring the inefficiency of decision making units. Eur J Oper Res. 1978;2(6):429–44.
Deprins D, Simar L, Tulkens H. Measuring labor inefficiency in post offices. In: Marchand M, Pestieau P, Tulkens H, editors. The performance of public enterprises: concepts and measurements. Amsterdam: North-Holland; 1984. p. 243–67.
Efron B. Bootstrap methods: another look at the jackknife. Ann Stat. 1979;7:1–16.
Efron B. The Jackknife, the Bootstrap and other Resampling plans, CBMS-NSF regional conference series in applied mathematics, #38. Philadelphia: SIAM; 1982.
Efron B, Tibshirani RJ. An introduction to the Bootstrap. London: Chapman and Hall; 1993.
Färe R. Fundamentals of production theory. Berlin: Springer; 1988.
Färe R, Grosskopf S, Lovell CAK. The measurement of efficiency of production. Boston: Kluwer-Nijhoff; 1985.
Farrell MJ. The measurement of productive efficiency. J Roy Stat Soc A. 1957;120:253–81.
Gijbels I, Mammen E, Park BU, Simar L. On estimation of monotone and concave frontier functions. J Am Stat Assoc. 1999;94:220–8.
Hall P. The Bootstrap and Edgeworth expansion. New York, NY: Springer; 1992.
Hall P, Härdle W, Simar L. Iterated bootstrap with application to frontier models. J Product Anal. 1995;6:63–76.
Jeong SO, Simar L. Linearly interpolated FDH efficiency score for nonconvex frontiers. J Multivar Anal. 2006;97:2141–61.
Kneip A, Park BU, Simar L. A note on the convergence of nonparametric DEA estimators for production efficiency scores. Economet Theor. 1998;14:783–93.
Kneip A, Simar L, Wilson PW. Asymptotics and consistent bootstraps for DEA estimators in non-parametric frontier models. Economet Theor. 2008;24:1663–97.
Kneip A, Simar L, Wilson PW. A computationally efficient, consistent bootstrap for inference with non-parametric DEA estimators, Computational Economics (2011, in press).
Korostelev A, Simar L, Tsybakov AB. Efficient estimation of monotone boundaries. Ann Stat. 1995a;23:476–89.
Korostelev A, Simar L, Tsybakov AB. On estimation of monotone and convex boundaries, Publications de l’Institut de Statistique des Universités de Paris XXXIX 1. 1995b. 3–18.
Lewis PA, Goodman AS, Miller JM. A pseudo-random number generator for the System/360. IBM Syst J. 1969;8:136–46.
Löthgren M. How to Bootstrap DEA Estimators: A Monte Carlo Comparison (contributed paper presented at the Third Biennal Georgia Productivity Workshop, University of Georgia, Athens, GA, October 1998), Working paper series in economics and Finance #223, Department of Economic Statistics, Stockhold School of Economics, Sweden. 1998.
Löthgren M. Bootstrapping the Malmquist productivity index-A simulation study. Appl Econ Lett. 1999;6:707–10.
Löthgren M, Tambour M. Bootstrapping the DEA-based Malmquist productivity index, in essays on performance measurement in health care, Ph.D. dissertation by Magnus Tambour. Stockholm, Sweden: Stockholm School of Economics; 1997.
Löthgren M, Tambour M. Testing scale efficiency in DEA models: a bootstrapping approach. Appl Econ. 1999;31:1231–7.
Manski CF. Analog estimation methods in econometrics. New York: Chapman and Hall; 1988.
Park B, Simar L, Weiner C. The FDH estimator for productivity efficiency scores: asymptotic properties. Economet Theor. 1999;16:855–77.
Park BU, Jeong S-O, Simar L. Asymptotic distribution of conical-hull estimators of directional edges. Ann Stat. 2010;38:1320–40.
Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes. Cambridge: Cambridge University Press; 1986.
Scott DW. Multivariate density estimation. New York, NY: Wiley; 1992.
Shephard RW. Theory of cost and production function. Princeton: Princeton University Press; 1970.
Silverman BW. Choosing the window width when estimating a density. Biometrika. 1978;65:1–11.
Silverman BW. Density estimation for statistics and data analysis. London: Chapman & Hall Ltd.; 1986.
Simar L, Wilson PW. Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Manag Sci. 1998;44(11):49–61.
Simar L, Wilson PW. Some problems with the Ferrier/Hirschberg bootstrap idea. J Product Anal. 1999a;11:67–80.
Simar L, Wilson PW. Of course we can bootstrap DEA scores! But does it mean anything? Logic trumps wishful thinking. J Product Anal. 1999b;11:93–7.
Simar L, Wilson PW. Estimating and bootstrapping Malmquist indices. Eur J Oper Res. 1999c;115:459–71.
Simar L, Wilson PW. Statistical inference in nonparametric frontier models: the state of the art. J Product Anal. 2000a;13:49–78.
Simar L, Wilson PW. A general methodology for bootstrapping in nonparametric frontier models. J Appl Stat. 2000b;27:779–802.
Simar L, Wilson PW. Nonparametric tests of returns to scale. Eur J Oper Res. 2002;139:115–32.
Simar L, Wilson PW. Statistical inference in nonparametric frontier models: Recent developments and perspectives. In: Fried H, Lovell CAK, Schmidt S, editors. The measurement of productive efficiency, chapter 4. 2nd ed. Oxford: Oxford University Press; 2008. p. 421–521.
Simar L, Wilson PW. 2009 Inference by subsampling in nonparametric frontier models. Discussion paper #0933, Institut de Statistique, Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium.
Spanos A. Statistical foundations of econometric modelling. Cambridge: Cambridge University Press; 1986.
Swanepoel JWH. A note on proving that the (modified) bootstrap works. Comm Stat Theor Meth. 1986;15:3193–203.
Wheelock DC, Wilson PW. Explaining bank failures: deposit insurance, regulation, and efficiency. Rev Econ Stat. 1995;77:689–700.
Wheelock DC, Wilson PW. Why do banks disappear? The determinants of US bank failures and acquisitions. Rev Econ Stat. 2000;82:127–38.
Wilson PW. Testing independence in models of productive efficiency. J Prod Anal. 2003;20:361–90.
Wilson PW. FEAR: a software package for frontier efficiency analysis with R. Soc Econ Plann Sci. 2008;42:247–54.
Wilson PW. Asymptotic properties of some non-parametric hyperbolic efficiency estimators. In: van Keilegom I, Wilson PW, editors. Exploring research frontiers in contemporary statistics and econometrics. Berlin: Physica-Verlag; 2010.
Acknowledgments
Léopold Simar gratefully acknowledges the Research support from “Projet d’Actions de Recherche Concertées” (No. 98/03-217) and from the “Inter-university Attraction Pole,” Phase V (No. P5/24) from the Belgian Government.
Paul W. Wilson also gratefully acknowledges Research support from the Texas Advanced Computing Center at the University of Texas, Austin.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Simar, L., Wilson, P.W. (2011). Performance of the Bootstrap for DEA Estimators and Iterating the Principle. In: Cooper, W., Seiford, L., Zhu, J. (eds) Handbook on Data Envelopment Analysis. International Series in Operations Research & Management Science, vol 164. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6151-8_10
Download citation
DOI: https://doi.org/10.1007/978-1-4419-6151-8_10
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-6150-1
Online ISBN: 978-1-4419-6151-8
eBook Packages: Business and EconomicsBusiness and Management (R0)