Advertisement

Statistics and Computing

, Volume 6, Issue 2, pp 147–157 | Cite as

Control variates and importance sampling for efficient bootstrap simulations

  • Tim Hesterberg
Papers

Abstract

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.

We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n-1/2B-1, where n is the sample size and B is the bootstrap sample size.

We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.

Keywords

Variance reduction control variates bootstrap saddlepoint 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth and Brooks/Cole, Pacific Grove, CA.Google Scholar
  2. Booth, J. G. and Hall, P. (1994) Monte Carlo approximation and the iterated bootstrap. Biometrika, 81(2), 331.Google Scholar
  3. Booth, J. G., Hall, P. and Wood, Andrew T. A. (1992) Balanced importance resampling for the bootstrap. Annals of Statistics.Google Scholar
  4. Breiman, L. and Friedman, J. H. (1985) Estimating optimal transformations for multiple regression and correlation (with discussion). Journal of the American Statistical Association, 80, 580–619.Google Scholar
  5. Daniels, H. E. (1987) Tail probability approximations. International Statistical Review, 55(1), 37–48.Google Scholar
  6. Daniels, H. E. and Young, G. A. (1991) Saddlepoint approximation for the studentized mean, with an application to the bootstrap. Biometrika, 78(1), 169–79.Google Scholar
  7. Davison, A. C. (1988) Discussion of paper by D. V. Hinkley. Journal of the Royal Statistical Society, Series B, 50, 356–7.Google Scholar
  8. Davison, A. C. and Hinkley, D. V. (1988) Saddlepoint approximations in resampling methods. Biometrika, 75, 417–31.Google Scholar
  9. Davison, A. C., Hinkley, D. V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 74, 555–66.Google Scholar
  10. DiCiccio, T. J. and Tibshirani, R. (1987) Bootstrap confidence intervals and bootstrap approximations. Journal of the American Statistical Society, 82(397), 163–70.Google Scholar
  11. DiCiccio, T. J., Martin, M. A. and Young, G. A. (1994) Analytical approximations to bootstrap distributions functions using Saddlepoint methods. Statistica Sinica, 4(1), 281.Google Scholar
  12. Do, K. (1992) A simulation study of balanced and antithetic bootstrap resampling methods. Journal of Statistical Computing and Simulation, 40, 153–66.Google Scholar
  13. Do, K. and Hall, P. (1991a) On importance resampling for the bootstrap. Biometrika, 78(1), 161–7.Google Scholar
  14. Do, K. and Hall, P. (1991b) Quasi-random sampling for the bootstrap. Statistics and Computing, 1(1), 13–22.Google Scholar
  15. Do, K. and Hall, P. (1991b) Distribution estimation using concomitants of order statistics, with application to Monte Carlo simulations for the bootstrap. Journal of the Royal Statistical Society, Series B, 54(2), 595–607.Google Scholar
  16. Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
  17. Efron, B. (1987) Better bootstrap confidence intervals (with discussion). Journal of the American Statistical Association, 82(397), 171–200.Google Scholar
  18. Efron, B. (1990) More efficient bootstrap computations. Journal of the American Statistical Association, 85, 79–89.Google Scholar
  19. Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. Chapman & Hall, London.Google Scholar
  20. Gleason, J. R. (1988) Algorithms for balanced bootstrap simulations. American Statistician, 42, 263–6.Google Scholar
  21. 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, Series B, 52, 185–202.Google Scholar
  22. Hall, P. (1989a) On efficient bootstrap simulation. Biometrika, 76, 613–7.Google Scholar
  23. Hall, P. (1989b) Antithetic resampling for the bootstrap. Biometrika, 76, 713–24.Google Scholar
  24. Hammersley, J. M. and Hanscomb, D. C. (1964) Monte Carlo Methods. Methuen, London.Google Scholar
  25. Hesterberg, T. C. (1987) Importance sampling in multivariate problems. Proceedings of the Statistical Computing Section, American Statistical Association 1987 Meeting, pp. 412–417.Google Scholar
  26. Hesterberg, T. C. (1988) Advances in Importance Sampling. Ph.D. dissertation, Statistics Department, Stanford University.Google Scholar
  27. Hesterberg, T. C. (1994) Saddlepoint quantiles and distribution curves, with bootstrap applications. Computational Statistics, 9(3), 207–12.Google Scholar
  28. Hesterberg, T. C. (1995a) Weighted average importance sampling and defensive mixture distributions. Technometrics, 37 (2), 185–94.Google Scholar
  29. Hesterberg, T. C. (1995b) Tail-specific linear approximations for efficient bootstrap simulations. Journal of Computational and Graphical Statistics, 4(2), 113–33.Google Scholar
  30. Hsu, J. C. and Nelson, B. L. (1990) Control variates for quantile estimation. Management Science, 36, 835–51.Google Scholar
  31. Hinkley, D. V. and Shi, S. (1989) Importance sampling and the nested bootstrap. Biometrika, 76(3) 435–46.Google Scholar
  32. Johns, M. V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83(403), 709–14.Google Scholar
  33. Larsen, R. J. and Marx, M. L. (1986) An Introduction to Mathematical Statistics and Its Applications. Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
  34. Lavenberg, S. S. and Welch, P. D. (1991) A perspective on the use of control variables to increase the efficiency of Monte Carlo simulations. Management Science, 27(3), pp322–35.Google Scholar
  35. Statistical Sciences, Inc. (1991) S-PLUS Reference Manual, Version 3.0. Statistical Sciences, Inc., Seattle.Google Scholar
  36. Therneau, T. M. (1983) Variance reduction techniques for the bootstrap. Technical Report No. 200 (Ph.D. Thesis), Department of Statistics, Stanford University.Google Scholar
  37. Tibshirani, R. J. (1984) Bootstrap confidence intervals. Technical Report LCS-3, Department of Statistics, Stanford University.Google Scholar

Copyright information

© Chapman & Hall 1996

Authors and Affiliations

  • Tim Hesterberg
    • 1
  1. 1.Mathematics DepartmentFranklin & Marshall CollegeLancasterUSA

Personalised recommendations