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 -1/2 B -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.
Similar content being viewed by others
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth and Brooks/Cole, Pacific Grove, CA.
Booth, J. G. and Hall, P. (1994) Monte Carlo approximation and the iterated bootstrap. Biometrika, 81(2), 331.
Booth, J. G., Hall, P. and Wood, Andrew T. A. (1992) Balanced importance resampling for the bootstrap. Annals of Statistics.
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.
Daniels, H. E. (1987) Tail probability approximations. International Statistical Review, 55(1), 37–48.
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.
Davison, A. C. (1988) Discussion of paper by D. V. Hinkley. Journal of the Royal Statistical Society, Series B, 50, 356–7.
Davison, A. C. and Hinkley, D. V. (1988) Saddlepoint approximations in resampling methods. Biometrika, 75, 417–31.
Davison, A. C., Hinkley, D. V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 74, 555–66.
DiCiccio, T. J. and Tibshirani, R. (1987) Bootstrap confidence intervals and bootstrap approximations. Journal of the American Statistical Society, 82(397), 163–70.
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.
Do, K. (1992) A simulation study of balanced and antithetic bootstrap resampling methods. Journal of Statistical Computing and Simulation, 40, 153–66.
Do, K. and Hall, P. (1991a) On importance resampling for the bootstrap. Biometrika, 78(1), 161–7.
Do, K. and Hall, P. (1991b) Quasi-random sampling for the bootstrap. Statistics and Computing, 1(1), 13–22.
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.
Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia.
Efron, B. (1987) Better bootstrap confidence intervals (with discussion). Journal of the American Statistical Association, 82(397), 171–200.
Efron, B. (1990) More efficient bootstrap computations. Journal of the American Statistical Association, 85, 79–89.
Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. Chapman & Hall, London.
Gleason, J. R. (1988) Algorithms for balanced bootstrap simulations. American Statistician, 42, 263–6.
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.
Hall, P. (1989a) On efficient bootstrap simulation. Biometrika, 76, 613–7.
Hall, P. (1989b) Antithetic resampling for the bootstrap. Biometrika, 76, 713–24.
Hammersley, J. M. and Hanscomb, D. C. (1964) Monte Carlo Methods. Methuen, London.
Hesterberg, T. C. (1987) Importance sampling in multivariate problems. Proceedings of the Statistical Computing Section, American Statistical Association 1987 Meeting, pp. 412–417.
Hesterberg, T. C. (1988) Advances in Importance Sampling. Ph.D. dissertation, Statistics Department, Stanford University.
Hesterberg, T. C. (1994) Saddlepoint quantiles and distribution curves, with bootstrap applications. Computational Statistics, 9(3), 207–12.
Hesterberg, T. C. (1995a) Weighted average importance sampling and defensive mixture distributions. Technometrics, 37 (2), 185–94.
Hesterberg, T. C. (1995b) Tail-specific linear approximations for efficient bootstrap simulations. Journal of Computational and Graphical Statistics, 4(2), 113–33.
Hsu, J. C. and Nelson, B. L. (1990) Control variates for quantile estimation. Management Science, 36, 835–51.
Hinkley, D. V. and Shi, S. (1989) Importance sampling and the nested bootstrap. Biometrika, 76(3) 435–46.
Johns, M. V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83(403), 709–14.
Larsen, R. J. and Marx, M. L. (1986) An Introduction to Mathematical Statistics and Its Applications. Prentice-Hall, Englewood Cliffs, New Jersey.
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.
Statistical Sciences, Inc. (1991) S-PLUS Reference Manual, Version 3.0. Statistical Sciences, Inc., Seattle.
Therneau, T. M. (1983) Variance reduction techniques for the bootstrap. Technical Report No. 200 (Ph.D. Thesis), Department of Statistics, Stanford University.
Tibshirani, R. J. (1984) Bootstrap confidence intervals. Technical Report LCS-3, Department of Statistics, Stanford University.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hesterberg, T. Control variates and importance sampling for efficient bootstrap simulations. Stat Comput 6, 147–157 (1996). https://doi.org/10.1007/BF00162526
Issue Date:
DOI: https://doi.org/10.1007/BF00162526