On the Double Bootstrap

  • Michael A. Martin


We present a discussion of iterated bootstrap procedures for the problems of bias reduction in point estimation and the reduction of coverage error in confidence intervals. In the case of coverage correction in confidence intervals, bootstrap iteration is directed at obtaining intervals with high coverage accuracy. We consider several other properties of these intervals and assess some of the practical advantages and disadvantages of the double bootstrap technique.


Bootstrap Estimate Bootstrap Confidence Interval Edgeworth Expansion Coverage Accuracy Bias Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramovitch, L. and Singh, K. (1985), Edge worth corrected pivotal statistics and the bootstrap, Ann. Statist, 13, 116–132.MathSciNetMATHCrossRefGoogle Scholar
  2. Beran, R. (1987), Prepivoting to reduce level error in confidence sets, Biometrika, 74, 457–468.MathSciNetMATHCrossRefGoogle Scholar
  3. Beran, R. (1988), Prepivoting test statistics: a bootstrap view of asymptotic refinements, J. Amer. Statist. Assoc, 83, 687–697.MathSciNetMATHCrossRefGoogle Scholar
  4. Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986), Efficient bootstrap simulation, Biometrika, 73, 555–556.MathSciNetMATHCrossRefGoogle Scholar
  5. DiCiccio, T.J. and Romano, J.P. (1988), A review of bootstrap confidence intervals, (With discussion,) J. Roy. Statist. Soc. B, 50, 338–370.MathSciNetMATHGoogle Scholar
  6. Efron, B. (1979), Bootstrap methods: another look at the jackknife, Ann. Statist., 7, 1–26.MathSciNetMATHCrossRefGoogle Scholar
  7. Efron, B. (1981), Nonparametric standard errors and confidence intervals: the jackknife, the bootstrap and other methods, Canadian J. Statist., 9, 139–172.MathSciNetMATHCrossRefGoogle Scholar
  8. Efron, B. (1982), The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia.CrossRefGoogle Scholar
  9. Efron, B. (1983), Estimating the error rate of a prediction rule: Improvement on cross-validation, Biometrika, 72, 45–58.MathSciNetCrossRefGoogle Scholar
  10. Efron, B. (1987), Better bootstrap confidence intervals, (With discussion,) J. Amer. Statist. Assoc, 82, 171–200.MathSciNetMATHCrossRefGoogle Scholar
  11. Efron, B. (1988), Bootstrap confidence intervals: good or bad? (With discussion.) Psych. Bull, 104, 293–296.CrossRefGoogle Scholar
  12. Efron, B. (1990), More efficient bootstrap computations, J. Amer. Statist. Assoc, 85, 79–89.MathSciNetMATHCrossRefGoogle Scholar
  13. Hall, P. (1986), On the bootstrap and confidence intervals, Ann. Statist, 14, 1431–1452.MathSciNetMATHCrossRefGoogle Scholar
  14. Hall, P. (1988), Theoretical comparisons of bootstrap confidence intervals, (With discussion,) Ann. Statist., 16, 927–985.MathSciNetMATHCrossRefGoogle Scholar
  15. Hall, P. (1989), On efficient bootstrap simulation, Biometrika, 76, 613–617.MathSciNetMATHCrossRefGoogle Scholar
  16. Hall, P. and Martin, M.A. (1988), On bootstrap resampling and iteration, Biometrika, 75, 661–671.MathSciNetMATHCrossRefGoogle Scholar
  17. Hall, P., Martin, M.A. and Schucany, W.R. (1989), Better nonparametric bootstrap confidence intervals for the correlation coefficient, J. Statist. Comput. Simul, 33, 161–172.MathSciNetMATHCrossRefGoogle Scholar
  18. Hinkley, D.V. (1988), Bootstrap methods, (With discussion.) J. Roy. Statist. Soc B, 50, 321–337.MathSciNetMATHGoogle Scholar
  19. Hinkley, D.V. and Shi, S. (1989), Importance sampling and the nested bootstrap, Biometrika, 76, 435–446.MathSciNetMATHCrossRefGoogle Scholar
  20. Johns, M.V. (1988) Importance sampling for bootstrap confidence intervals, J. Amer. Statist. Assoc, 83, 709–714.MathSciNetMATHCrossRefGoogle Scholar
  21. Loh, W.-Y. (1987), Calibrating confidence coefficients, J. Amer. Statist. Assoc, 82, 155–162.MathSciNetMATHCrossRefGoogle Scholar
  22. Martin, M.A. (1990), On bootstrap iteration for coverage correction in confidence intervals, J. Amer. Statist Assoc, to appear.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Michael A. Martin
    • 1
  1. 1.Department of StatisticsStanford UniversityStanfordUSA

Personalised recommendations