Abstract
A simple yet easy to implement method is proposed to further improve the finite sample approximation of the recently developed central limit theorems for aggregates of envelopment estimators. Focusing on the simple mean efficiency, we propose using the bias-corrected individual efficiency estimate to improve the variance estimator. The extensive Monte-Carlo experiments confirm that, for relatively small sample sizes (≤100), with both low dimensions and especially for high dimensions, our new method combined with the data sharpening method generally provides better ‘coverage’ (of the true values by the estimated confidence intervals) than the previously developed approaches.
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Notes
For more details about the simulation process, see the Appendix A.
The data sharpening method works by smoothing-out the estimated efficiency scores that are close to 1, i.e., those smaller than 1 + τ, where τ = n−γ and γ is in the range of [κ/2, κ]. The idea is to use a uniform density with the range [1, 1 + τ] to approximate the density of the efficiency locally in the τ neighborhood of the frontier. Further, the results from simulations in Nguyen et al. (2022) indicate the level of γ near 0.75κ seems to perform best.
For the values of the empirical coverages, see Table A.2 in the separate Appendix A. Further, when κ = 2/5 for VRS-DEA, Kneip et al. (2015) finds that both Theorem 4.3 and Theorem 4.4 in Kneip et al. (2015) are applicable, while Theorem 4.4 is preferred. Thus, for p = 3 and q = 1, we only include the results from the preferred method.
The simulation results for CRS-DEA estimators are summarized in Figure C.1 of the separate Appendix C, which also confirm the improved performance of our proposed method.
We thank an anonymous referee for encouraging exploring this avenue.
For more details, see the separate Appendix B.
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Acknowledgements
VZ acknowledges the support from the Australian Research Council (FT170100401) and The University of Queensland. SZ acknowledges the support from Liaoning Social Science Foundation (L22CJY011) and Liaoning Provincial Department of Education Research Fund (LJKMZ20221602). Feedback from Bao Hoang Nguyen, Evelyn Smart and Zhichao Wang is appreciated. These individuals and organizations are not responsible for the views expressed here.
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Simar, L., Zelenyuk, V. & Zhao, S. Further improvements of finite sample approximation of central limit theorems for envelopment estimators. J Prod Anal 59, 189–194 (2023). https://doi.org/10.1007/s11123-023-00661-8
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DOI: https://doi.org/10.1007/s11123-023-00661-8