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Inference by the m out of n bootstrap in nonparametric frontier models

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Abstract

It is well-known that the naive bootstrap yields inconsistent inference in the context of data envelopment analysis (DEA) or free disposal hull (FDH) estimators in nonparametric frontier models. For inference about efficiency of a single, fixed point, drawing bootstrap pseudo-samples of size m < n provides consistent inference, although coverages are quite sensitive to the choice of subsample size m. We provide a probabilistic framework in which these methods are shown to valid for statistics comprised of functions of DEA or FDH estimators. We examine a simple, data-based rule for selecting m suggested by Politis et al. (Stat Sin 11:1105–1124, 2001), and provide Monte Carlo evidence on the size and power of our tests. Our methods (i) allow for heterogeneity in the inefficiency process, and unlike previous methods, (ii) do not require multivariate kernel smoothing, and (iii) avoid the need for solutions of intermediate linear programs.

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Notes

  1. FDH estimators are also widely used, though not as widely as DEA estimators. On 11 August 2010, a search with Google Scholar using the keywords “fdh”, “input”, “output”, and “efficiency” returned approximately 2,080 results. Statistical properties of FDH estimators have been examined by Korostelev et al. (1995a), Park et al. (2000), Wheelock and Wilson (2008), and Wilson (2010).

  2. Convexity of the production set is typically assumed in the microeconomic theory of the firm for convenience. However, there may be cases where the assumption is violated, and there is no theorem saying that production sets are necessarily convex.

  3. By “nondegenerate,” we mean the distribution is not a probability mass at a single point.

  4. In passing from Eqs. 31 to 33 we have implicitly assumed uniform integrability of \((W_n(\user2{Z}))^{2+\epsilon}\) for some ε > 0, i.e., \(\lim_{n\to\infty}\sup E\left(|W_n(\user2{Z})|^{2+\epsilon}\right)<\infty\) for some ε > 0. See Serfling (1980) or van der Vaart (2000) for details.

  5. The argument used here is standard; in addition, it is straightforward to verify that the result also holds in the unrestricted case where \({P\in{\mathbb P}}\). The only difference will be in the expressions for the mean and the variance of \({\tau_n({\mathcal S}_n)}\) that appear in Eqs. 35 and 36. Of course, to satisfy the Lyapunov condition an additional technical regularity condition on the random variable \(T(\user2{Z})\) is needed. In particular, \(T(\user2{Z})\) must have finite moments up to order 4 (when H 0 is true, \({P\in{\mathbb P}_0}\) and \(T(\user2{Z})\) is a degenerate random variable equal to zero). Hence, for \({P\in {\mathbb P}}\),

    $$ \sqrt{n} \left(\tau_n({\mathcal S}_n) -\left(\tau(P)+ \mu_Q/n^{\kappa}\right)\right) {\mathop{\longrightarrow}\limits^{\mathcal L}}\, {\mathcal N} \left(0,\sigma^2_Q/n^{2\alpha} + O\left(n^{-\kappa}\right)\sigma(P) +\sigma^2(P) \right). $$

    Note that the rate of convergence is slower when the null is false.

  6. Of course, situations involving more than one output can be easily handled using our methods; here, we use only one output to simplify the process of simulating data. In all of the theoretical results about properties of DEA and FDH estimators, including Korostelev et al. (1995a, b), Park et al. (2000, 2010), Kneip et al. (1998, 2008, 2010), Gijbels et al. (1999), and Wilson (2010), it is the dimensionality (p + q) rather than the ratio p/q that is important for determining properties of the estimators; consequently, we expect no loss of generality from simulating only one output.

  7. The interval given by Eq. 19 is a special case of this, where \({\widehat \theta}\) is the VRS-DEA estimator.

  8. Results from the experiments using the m-bootstrap are available in a separate appendix, available on-line at http://www.stat.ucl.ac.be/ISpub/ISdp.html/.

  9. Figure A.1 in the separate appendix mentioned in footnote 8 shows plots of rejection rates versus subsample sizes analogous to those in Fig. 2, except that the rejection rates depicted in Fig. A.1 are those obtained using the m-bootstrap. Comparing the results shown in the two figures, it is apparent that for given dimensionality (p + q) and sample size n, the optimal subsample size m is smaller when resampling is done with replacement as opposed to without replacement. In addition, holding dimensionality and sample size constant, resampling with replacement typically results in less test power than resampling without replacement for given subsample sizes and departures from the null.

  10. Results for tests of convexity based on the statistic \({{\widehat \tau}_1({\mathcal S}_n)}\) defined in Eq. 42 are available in the separate appendix mentioned in footnote 8.

  11. Of course, the null hypothesis in our test is a composite hypothesis. We have considered only two values of δ where the null is true while the size of the test is equal to the supremum of rejection rates for each value of δ ∈ (0,1].

  12. Trials represented in Fig. 3 were chosen by generating uniform random integers between 1 and 1,024 (inclusive).

  13. Results for convexity tests based on the LFDH estimator are available in the separate appendix mentioned in footnote 8.

  14. Similar results from experiments using the m-bootstrap appear in Tables A.7–A.8 of the separate appendix.

  15. In our experiments with unbalanced resampling, setting the bootstrap efficiency estimator equal to one when the estimate is infeasible amounts to recognizing that the particular bootstrap replication has no useful information for inference, and avoids imposing conditioning in the bootstrap world that is not present in the real world. This does not alter the asymptotic properties of our bootstrap. A similar device was used by Jeong and Simar (2006).

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Acknowledgments

We have benefited from discussions with Peter Bickel, Alois Kneip, and Jean-Marie Rolin. Financial support from the “Interuniversity Attraction Pole”, Phase VI (No. P6/03) from the Belgian Government (Belgian Science Policy) and from the Chair of Excellency “Pierre de Fermat”, Région Midi-Pyrénées, France are gratefully acknowledged. This work was made possible by the Palmetto cluster maintained by Clemson Computing and Information Technology (CCIT) at Clemson University; we are grateful for technical support by the staff of CCIT. The usual caveats apply.

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Authors and Affiliations

  1. Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, 1348, Louvain-la-Neuve, Belgium

    Léopold Simar

  2. Toulouse School of Economics, IDEI, 31000, Toulouse, France

    Léopold Simar

  3. The John E. Walker Department of Economics, Clemson University, 222 Sirrine Hall, Clemson, SC, 29634-1309, USA

    Paul W. Wilson

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  1. Léopold Simar
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Correspondence to Paul W. Wilson.

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Simar, L., Wilson, P.W. Inference by the m out of n bootstrap in nonparametric frontier models. J Prod Anal 36, 33–53 (2011). https://doi.org/10.1007/s11123-010-0200-4

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