Brazilian Journal of Probability and Statistics

On improved estimation for importance sampling

David Firth

Full-text: Open access


The standard estimator used in conjunction with importance sampling in Monte Carlo integration is unbiased but inefficient. An alternative estimator is discussed, based on the idea of a difference estimator, which is asymptotically optimal. The improved estimator uses the importance weight as a control variate, as previously studied by Hesterberg (Ph.D. Dissertation, Stanford University (1988); Technometrics 37 (1995) 185–194; Statistics and Computing 6 (1996) 147–157); it is routinely available and can deliver substantial additional variance reduction. Finite-sample performance is illustrated in a sequential testing example. Connections are made with methods from the survey-sampling literature.

Article information

Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 437-443.

First available in Project Euclid: 22 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Difference estimator Horvitz–Thompson estimator regression estimator simulation variance reduction


Firth, David. On improved estimation for importance sampling. Braz. J. Probab. Stat. 25 (2011), no. 3, 437--443. doi:10.1214/11-BJPS155.

Export citation


  • Evans, M. and Swartz, T. B. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford: Oxford University Press.
  • Firth, D. and Bennett, K. E. (1998). Robust models in probability sampling (with discussion). Journal of the Royal Statistical Society B 60 3–21.
  • Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods. London: Chapman and Hall.
  • Hesterberg, T. C. (1988). Advances in importance sampling. Ph.D. thesis, Stanford Univ.
  • Hesterberg, T. C. (1995). Weighted average importance sampling and defensive mixture distributions. Technometrics 37 185–194.
  • Hesterberg, T. C. (1996). Control variates and importance sampling for efficient bootstrap simulations. Statistics and Computing 6 147–157.
  • Ripley, B. D. (1987). Stochastic Simulation. New York: Wiley.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. New York: Springer.
  • Särndal, C. E., Swensson, B. and Wretman, J. H. (1992). Model Assisted Survey Sampling. New York: Springer.
  • Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Annals of Statistics 25 673–684.
  • Van Deusen, P. C. (1995). Difference sampling as an alternative to importance sampling. Canadian Journal of Forest Research 25 487–490.