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.
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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.
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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.
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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
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