Bootstrap sample size in nonregular cases

Author:
Jun Shao

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

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**L. Abramovitch and K. Singh,*Edgeworth corrected pivotal statistics and the bootstrap*, Ann. Statist.**13**(1981), 116-132. MR**773156 (86f:62031)****[2]**K. B. Athreya,*Bootstrap of the mean in the infinite variance case*, Ann. Statist.**15**(1987), 724-731. MR**888436 (88g:62082)****[3]**G. J. Babu,*Bootstrapping statistics with linear combinations of chi-squares as weak limit*, Sankhyā Ser. A**46**(1984), 85-93. MR**768919 (87a:62048)****[4]**R. Beran,*Prepivoting to reduce level error of confidence sets*, Biometrika**74**(1987), 457-468. MR**909351 (89h:62049)****[5]**P. J. Bickel and D. A. Freedman,*Some asymptotic theory for the bootstrap*, Ann. Statist.**9**(1981), 1196-1217. MR**630103 (83a:62051)****[6]**P. Deheuvels, D. M. Mason, and G. R. Shorack,*Some results on the influence of extremes on the bootstrap*, Ann. Inst. Henri Poincaré**29**(1993), 83-103. MR**1204519 (94a:62068)****[7]**B. Efron,*Bootstrap methods*:*another look at the jackknife*, Ann. Statist.**7**(1979), 1-26. MR**515681 (80b:62021)****[8]**-,*The jackknife, the bootstrap, and other resampling plans*, SIAM, Philadelphia, 1982. MR**659849 (84a:62035)****[9]**-,*Better bootstrap confidence intervals*(*with discussion*), J. Amer. Statist. Assoc.**82**(1987), 171-200. MR**883345 (88m:62053)****[10]**P. Hall,*Theoretical comparison of bootstrap confidence intervals*(*with discussion*), Ann. Statist.**16**(1988), 927-985. MR**959185 (89h:62085)****[11]**-,*On the relative performance of bootstrap and Edgeworth approximations of a distribution function*, J. Multivariate Anal.**35**(1990), 108-129. MR**1084945 (91m:62077)****[12]**F. R. Hampel,*The influence curve and its role in robust estimation*, J. Amer. Statist. Assoc.**69**(1974), 383-393. MR**0362657 (50:15097)****[13]**J. S. Huang, P. K. Sen, and J. Shao,*Sample quantiles in some nonregular cases and the vulnerable bootstrap*, preprint, 1992.**[14]**W.-Y. Loh,*Estimating an endpoint of a distribution with resampling methods*, Ann. Statist.**12**(1984), 1543-1550. MR**760707 (86f:62066)****[15]**R. J. Serfling,*Approximation theorems of mathematical statistics*, Wiley, New York, 1980. MR**595165 (82a:62003)****[16]**J. Shao,*Bootstrap variable selection in regression*, preprint, 1992.**[17]**J. W. H. Swanepoel,*A note on proving that the (modified) bootstrap works*, Comm. Statist. A--Theory Methods**15**(1986), 3193-3203. MR**860478 (87m:62128)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1227529-8

Keywords:
Bootstrap sample size,
consistency,
differentiable functionals,
estimators based on tests,
functions of means

Article copyright:
© Copyright 1994
American Mathematical Society