Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Accuracy of the bootstrap approximation
Download PDF
Download PDF
  • Published: September 1991

Accuracy of the bootstrap approximation

  • Arup Bose1,2 &
  • G. J. Babu1,2 

Probability Theory and Related Fields volume 90, pages 301–316 (1991)Cite this article

  • 143 Accesses

  • 7 Citations

  • Metrics details

Summary

The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Abramovitch, L., Singh, K.: Edgeworth corrected pivotal statistics and the bootstrap. Ann. Stat.13, 116–132 (1985)

    Google Scholar 

  • Babu, G.J., Bai, Z.D.: Edgeworth expansions under minimal moment conditions. Tech. Report No. 90-17, Center for multivariate analysis, Department of Statistics, Pennsylvania State University University Park, PA, 16802, 1990

    Google Scholar 

  • Babu, G.J., Bose, A.: Bootstrap confidence intervals. Stat. Probab. Lett.7, 151–160 (1988)

    Google Scholar 

  • Babu, G.J., Singh, K.: Onr-quick limit sets for empirical and related processes based on mixing random variables. J. Multivariate Anal.12, 508–525 (1982)

    Google Scholar 

  • Babu, G.J., Singh, K.: On one term Edgeworth correction by Efron's bootstrap. Sankhyā, Ser. A46, 219–232 (1984)

    Google Scholar 

  • Bhattacharya, R.N., Qumsiyeh, M.: Second order andL p comparisons between the bootstrap and empirical Edgeworth expansion methodologies. Ann. Stat.17, (1) 160–169 (1989)

    Google Scholar 

  • Bhattacharya, R.N., Ranga Rao, R.: Normal approximation and asymptotic expansion. New York: Wiley 1976

    Google Scholar 

  • Bickel, P.J., Freedman, D.: Some asymptotics on the bootstrap. Ann. Stat.9, 1196–1217 (1981)

    Google Scholar 

  • Hall, P.: On the bootstrap and confidence intervals. Ann. Stat.14, 1431–1452 (1986)

    Google Scholar 

  • Hall, P.: Theoretical comparison of bootstrap confidence intervals (with discussion). Ann. Stat.16, (1) 927–985 (1988)

    Google Scholar 

  • Johnson, N.J.: Modifiedt-tests and confidence intervals for asymmetrical population. J. Am. Stat. Assoc.73, 536–544 (1978)

    Google Scholar 

  • Michel, R.: Non uniform central limit bounds with applications to probabilities of deviation. Ann. Probab.4, 102–106 (1976)

    Google Scholar 

  • Singh, K.: On asymptotic accuracy of Efron's bootstrap. Ann. Stat.9, 1187–1195 (1981)

    Google Scholar 

  • Sweeting, T.J.: Speeds of convergence for the multidimensional central limit theorem. Ann. Probab.5, 28–41 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Statistics and Mathematics Division, Indian Statistical Institute, 203 B.T. Road, 700035, Calcutta, India

    Arup Bose & G. J. Babu

  2. Department of Statistics, Pennsylvania State University, 219 Pond Laboratory, 16802, University Park, PA, USA

    Arup Bose & G. J. Babu

Authors
  1. Arup Bose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. J. Babu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSA Grant MDA 904-90-H-1001

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bose, A., Babu, G.J. Accuracy of the bootstrap approximation. Probab. Th. Rel. Fields 90, 301–316 (1991). https://doi.org/10.1007/BF01193748

Download citation

  • Received: 15 January 1990

  • Revised: 02 October 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01193748

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Confidence Interval
  • Normal Distribution
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature