Bootstrap sample size in nonregular cases
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- by Jun Shao PDF
- Proc. Amer. Math. Soc. 122 (1994), 1251-1262 Request permission
Abstract:
We study the bootstrap estimator of the sampling distribution of a given statistic in some nonregular cases where the given statistic is nonsmooth or not-so-smooth. It is found that the ordinary bootstrap, based on a bootstrap sample of the same size as the original data set, produces an inconsistent bootstrap estimator. On the other hand, when we draw a bootstrap sample of a smaller size with the ratio of the size of the bootstrap sample over the size of the original data set tending to zero, the resulting bootstrap estimator is consistent. Examples of these nonregular cases are given, including the cases of functions of means with null first-order derivatives, differentiable statistical functionals with null influence function, nondifferentiable functions of means, and estimators based on some test results.References
- Lavy Abramovitch and Kesar Singh, Edgeworth corrected pivotal statistics and the bootstrap, Ann. Statist. 13 (1985), no. 1, 116–132. MR 773156, DOI 10.1214/aos/1176346580
- K. B. Athreya, Bootstrap of the mean in the infinite variance case, Ann. Statist. 15 (1987), no. 2, 724–731. MR 888436, DOI 10.1214/aos/1176350371
- Gutti Jogesh Babu, Bootstrapping statistics with linear combinations of chi-squares as weak limit, Sankhyā Ser. A 46 (1984), no. 1, 85–93. MR 768919
- Rudolf Beran, Prepivoting to reduce level error of confidence sets, Biometrika 74 (1987), no. 3, 457–468. MR 909351, DOI 10.1093/biomet/74.3.457
- Peter J. Bickel and David A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), no. 6, 1196–1217. MR 630103
- Paul Deheuvels, David M. Mason, and Galen R. Shorack, Some results on the influence of extremes on the bootstrap, Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), no. 1, 83–103 (English, with English and French summaries). MR 1204519
- B. Efron, Bootstrap methods: another look at the jackknife, Ann. Statist. 7 (1979), no. 1, 1–26. MR 515681
- Bradley Efron, The jackknife, the bootstrap and other resampling plans, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1982. MR 659849
- Bradley Efron, Better bootstrap confidence intervals, J. Amer. Statist. Assoc. 82 (1987), no. 397, 171–200. With comments and a rejoinder by the author. MR 883345
- Peter Hall, Theoretical comparison of bootstrap confidence intervals, Ann. Statist. 16 (1988), no. 3, 927–985. With a discussion and a reply by the author. MR 959185, DOI 10.1214/aos/1176350933
- Peter Hall, On the relative performance of bootstrap and Edgeworth approximations of a distribution function, J. Multivariate Anal. 35 (1990), no. 1, 108–129. MR 1084945, DOI 10.1016/0047-259X(90)90019-E
- Frank R. Hampel, The influence curve and its role in robust estimation, J. Amer. Statist. Assoc. 69 (1974), 383–393. MR 362657 J. S. Huang, P. K. Sen, and J. Shao, Sample quantiles in some nonregular cases and the vulnerable bootstrap, preprint, 1992.
- Wei-Yin Loh, Estimating an endpoint of a distribution with resampling methods, Ann. Statist. 12 (1984), no. 4, 1543–1550. MR 760707, DOI 10.1214/aos/1176346811
- Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165 J. Shao, Bootstrap variable selection in regression, preprint, 1992.
- Jan W. H. Swanepoel, A note on proving that the (modified) bootstrap works, Comm. Statist. A—Theory Methods 15 (1986), no. 11, 3193–3203. MR 860478, DOI 10.1080/03610928608829303
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 1251-1262
- MSC: Primary 62G09
- DOI: https://doi.org/10.1090/S0002-9939-1994-1227529-8
- MathSciNet review: 1227529