Advertisement

Statistics and Computing

, Volume 2, Issue 3, pp 161–171 | Cite as

Analytical approximations for iterated bootstrap confidence intervals

  • Thomas J. DiCiccio
  • Michael A. Martin
  • G. Alastair Young
Papers

Abstract

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.

Keywords

Asymptotic approximations bootstrap calibration coverage accuracy resampling saddlepoint methods tail probability approximations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barndorff-Nielsen, O. E. and Cox, D. R. (1979) Edgeworth and saddlepoint approximations with statistical applications (with discussion).Journal of the Royal Statistical Society, B,41, 279–312.Google Scholar
  2. Beran, R. (1987) Prepivoting to reduce level error in confidence sets.Biometrika,74, 457–468.Google Scholar
  3. Beran, R. (1988) Prepivoting test statistics: a bootstrap view of asymptotic refinements.Journal of the American Statistical Association,83, 687–697.Google Scholar
  4. Daniels, H. E. and Young, G. A. (1991) Saddlepoint approximation for the Studentized mean, with an application to the bootstrap.Biometrika,78, 169–179.Google Scholar
  5. Davison, A. C. and Hinkley, D. V. (1988) Saddlepoint approximations in resampling methods.Biometrika,75, 417–431.Google Scholar
  6. DiCiccio, T. J. and Martin, M. A. (1991) Approximations of marginal tail probabilities for a class of smooth functions with applications to Bayesian and conditional inference.Biometrika,78, 891–902.Google Scholar
  7. DiCiccio, T. J., Martin, M. A. and Young, G. A. (1990) Analytic approximations to bootstrap distribution functions using saddlepoint methods. Technical Report No. 356, Department of Statistics, Stanford University.Google Scholar
  8. DiCiccio, T. J., Martin, M. A. and Young, G. A. (1991) Fast and accurate approximate double bootstrap confidence intervals. Technical Report No. 369, Department of Statistics, Stanford University.Google Scholar
  9. Efron, B. (1979) Bootstrap methods: another look at the jackknife.Annals of Statistics,7, 1–26.Google Scholar
  10. Efron, B. (1982)The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia.Google Scholar
  11. Efron, B. (1983) Estimating the error rate of a prediction rule: improvements in cross-validation.Journal of the American Statistical Association,78, 316–331.Google Scholar
  12. Efron, B. (1987) Better bootstrap confidence intervals (with discussion).Journal of the American Statistical Association,82, 171–185.Google Scholar
  13. Efron, B. (1992) Jackknife-after-bootstrap standard errors and influence functions (with discussion).Journal of the Royal Statistical Society, B,54, 83–127.Google Scholar
  14. Graham, R. L., Hinkley, D. V., John, P. W. M. and Shi, S. (1990) Balanced design of bootstrap simulations.Journal of the Royal Statistical Society B,52, 185–202.Google Scholar
  15. Hall, P. (1986) On the bootstrap and confidence intervals.Annals of Statistics,14, 1431–1452.Google Scholar
  16. Hall, P. (1988) Theoretical comparisons of bootstrap confidence intervals (with discussion).Annals of Statistics,16, 927–985.Google Scholar
  17. Hall, P. (1992) Efficient bootstrap simulation. InExploring the Limits of Bootstrap, ed. by Raoul LePage and Lynne Billard, pp. 127–143, Wiley, New York.Google Scholar
  18. Hall, P. and Martin, M. A. (1988) On bootstrap resampling and iteration.Biometrika,75, 661–671.Google Scholar
  19. Hall, P., Martin, M. A. and Schucany, W. R. (1989) Better nonparametric bootstrap confidence intervals for the correlation coefficient.Journal of Statistical Computation and Simulation,33, 161–172.Google Scholar
  20. Hinkley, D. V. and Shi, S. (1989) Importance sampling and the nested bootstrap.Biometrika,76, 435–446.Google Scholar
  21. Larsen, R. J. and Marx, M. L. (1990)Statistics. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  22. Loh, W.-Y. (1987) Calibrating confidence coefficients.Journal of the American Statistical Association,82, 155–162.Google Scholar
  23. Martin, M. A. (1990a) On bootstrap iteration for coverage correction in confidence intervals.Journal of the American Statistical Association,85, 1105–1118.Google Scholar
  24. Martin, M. A. (1990b) On the double bootstrap. InComputing Science and Statistics: Interface '90. Proceedings of the 22nd Symposium on the Interface, ed. Connie Page and Raoul LePage, pp. 73–78, Springer-Verlag, New York.Google Scholar
  25. Miller, R. G. Jr. (1986)Beyond Anova, Basics of Applied Statistics, Wiley, New York.Google Scholar
  26. Reid, N. (1988) Saddlepoint methods and statistical inference (with discussion).Statistical Science,3, 213–238.Google Scholar
  27. Rice, J. A. (1988)Mathematical Statistics and Data Analysis. Wadsworth and Brooks/Cole, Pacific Grove, CA.Google Scholar
  28. Schenker, N. (1985) Qualms about bootstrap confidence intervals.Journal of the American Statistical Association,80, 360–361.Google Scholar

Copyright information

© Chapman & Hall 1992

Authors and Affiliations

  • Thomas J. DiCiccio
    • 1
  • Michael A. Martin
    • 1
  • G. Alastair Young
    • 2
  1. 1.Department of StatisticsStanford UniversityStanfordUSA
  2. 2.Statistical LaboratoryUniversity of CambridgeCambridgeUK

Personalised recommendations