On a Set of the First Category

  • Hein Putter
  • Willem R. van Zwet

Abstract

In an analysis of the bootstrap Putter & van Zwet (1993) showed that under quite general circumstances, the bootstrap will work for “most” underlying distributions. In fact, the set of exceptional distributions for which the bootstrap does not work was shown to be a set D of the first category in the space S of all possible underlying distributions, equipped with a topology S. Such a set of the first category is usually “small” in a topological sense. However, it is known that this concept of smallness may sometimes be deceptive and in unpleasant cases such “small” sets may in fact be quite large.

Here we present a striking and hopefully amusing example of this phenomenon, where the “small” subset D equals all of S. We show that as a result, a particular version of the bootstrap for the sample minimum will never work, even though our earlier results tell us that it can only fail for a “small” subset of underlying distributions. We also show that when we change the topology on S—and as a consequence employ a different resampling distribution—this paradox vanishes and a satisfactory version of the bootstrap is obtained. This demonstrates the importance of a proper choice of the resampling distribution when using the bootstrap.

Keywords

Topo 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chang, L.-C. (1955), `On the ratio of the empirical distribution to the theoretical distribution function’, Acta Math. Sinica 5, 347–368. (English Translation in: Selected Translations in Mathematical Statististics and Probability, 4, 17–38 (1964).).MathSciNetMATHGoogle Scholar
  2. Devroye, L. & Györfi, L. (1990), `No empirical probability measure can converge in the total variation sense for all distributions’, Annals of Statistics 18, 1496–1499.MathSciNetMATHCrossRefGoogle Scholar
  3. Dudley, R. M. (1989), Real Analysis and Probability, Wadsworth, Belmont, California.MATHGoogle Scholar
  4. Le Cam, L. (1953), `On some asymptotic properties of maximum likelihood estimates and related Bayes estimates’, University of California Publications in Statistics 1, 277–330.Google Scholar
  5. Putter, H. (1994), Consistency of Resampling Methods, PhD thesis, University of Leiden.Google Scholar
  6. Putter, H. & van Zwet, W. R. (1993), Consistency of plug-in estimators with application to the bootstrap, Technical report, University of Leiden.Google Scholar
  7. Shorack, G. R. & Wellner, J. A. (1986), Empirical Processes with Applications to Statistics, Wiley, New York.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Hein Putter
    • 1
  • Willem R. van Zwet
    • 1
    • 2
  1. 1.University of LeidenHolland
  2. 2.University of North CarolinaUSA

Personalised recommendations