The Annals of Statistics

Resampling: consistency of substitution estimators

Hein Putter and Willem R. van Zwet

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On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional $\tau_N(P)$. An obvious and very general approach to this problem is to find an estimator $\hat{P}_N$ of P first, and then construct a so-called substitution estimator $\tau_N (\hat{P}_N)$ of $\tau_N(P)$. In this paper we investigate how to choose the estimator $\hat{P}_N$ so that the substitution estimator $\tau_N (\hat{P}_N)$ will be consistent.

Although our setup covers a broad class of estimation problems, the main substitution estimator we have in mind is a general version of the bootstrap where resampling is done from an estimated distribution $\hat{P}_N$. We do not focus in advance on a particular estimator $\hat{P}_N$, such as, for example, the empirical distribution, but try to indicate which resampling distribution should be used in a particular situation. The conclusion that we draw from the results and the examples in this paper is that the bootstrap is an exceptionally flexible method which comes into its own when full use is made of its flexibility. However, the choice of a good bootstrap method in a particular case requires rather precise information about the structure of the problem at hand. Unfortunately, this may not always be available.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2297-2318.

First available in Project Euclid: 16 September 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods
Secondary: 62F12: Asymptotic properties of estimators

Resampling bootstrap substitution estimator consistency set of first category equicontinuity local uniform convergence


Putter, Hein; van Zwet, Willem R. Resampling: consistency of substitution estimators. Ann. Statist. 24 (1996), no. 6, 2297--2318. doi:10.1214/aos/1032181156.

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  • BERAN, R. J. 1982. Estimated sampling distributions: the bootstrap and competitors. Ann. Statist. 10 212 225. Z.
  • BERAN, R. J. 1984. Bootstrap methods in statistics. Jber. Deutsch. Math.-Verein. 86 14 30. Z.
  • BICKEL, P. J. and FREEDMAN, D. A. 1981. Some asy mptotic theory for the bootstrap. Ann. Statist. 9 1196 1217. Z.
  • BILLINGSLEY, P. 1986. Convergence of Probability Measures. Wiley, New York. Z.
  • BIRGE, L. 1983. Approximation dans les espaces metriques et theorie de l'estimation. Z. ´ ´ ´ Wahrsch. Verw. Gebiete 65 181 237. Z.
  • BIRGE, L. 1986. On estimating a density using Hellinger distance and some other strange facts. ´ Probab. Theory Related Fields 71 271 291.
  • DUDLEY, R. M. 1989. Real Analy sis and Probability. Wadsworth, Belmont, CA. Z.
  • LE CAM, L. M. 1953. On some asy mptotic properties of maximum likelihood estimates and related Bay es' estimates. University of California Publications in Statistics 1 277 330. Z.
  • LE CAM, L. M. 1973. Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38 53. Z.
  • LE CAM, L. M. 1986. Asy mptotic Methods in Statistical Decision Theory. Springer, New York. Z.
  • LE CAM, L. M. and YANG, G. L. 1990. Asy mptotics in Statistics: Some Basic Concepts. Springer, New York. Z.
  • POLITIS, D. and ROMANO, J. 1994. Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031 2050. Z.
  • PUTTER, H. 1994. Consistency of resampling methods. Ph.D. thesis, Univ. Leiden. Z.
  • PUTTER, H. and VAN ZWET, W. R. 1996. On a set of the first category. In Festschrift for Lucien Le Cam. Springer, New York. To appear. Z.
  • VAN ZWET, W. R. 1996. Resampling: the jackknife and the naive bootstrap. Unpublished ¨ manuscript.