Electronic Journal of Statistics

Importance accelerated Robbins-Monro recursion with applications to parametric confidence limits

Zdravko I. Botev and Chris J. Lloyd

Full-text: Open access


Applying the standard stochastic approximation algorithm of Robbins and Monro (1951) to calculating confidence limits leads to poor efficiency and difficulties in estimating the appropriate governing constants as well as the standard error.

We suggest sampling instead from an alternative importance distribution and modifying the Robbins-Monro recursion accordingly. This can reduce the asymptotic variance by the usual importance sampling factor. It also allows the standard error and optimal step length to be estimated from the simulation. The methodology is applied to computing almost exact confidence limits in a generalised linear model.

Article information

Electron. J. Statist., Volume 9, Number 2 (2015), 2058-2075.

Received: November 2014
First available in Project Euclid: 17 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions
Secondary: 65C05: Monte Carlo methods

Stochastic approximation generalized linear model confidence limits profile upper limits importance sampling


Botev, Zdravko I.; Lloyd, Chris J. Importance accelerated Robbins-Monro recursion with applications to parametric confidence limits. Electron. J. Statist. 9 (2015), no. 2, 2058--2075. doi:10.1214/15-EJS1071. https://projecteuclid.org/euclid.ejs/1442513544

Export citation


  • Bardou, O., Frikha, N., and Páges, G.: Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling., Monte Carlo Methods and Applications, 15 (3): 173–210, 2009.
  • Blum, J. R.: Multidimensional stochastic approximation methods., The Annals of Mathematical Statistics, 25 (4): 737–744, 1954.
  • Brazzale, A. R., Davison, A. C., and Reid, N.:, Applied asymptotics: case studies in small-sample statistics, volume 23. Cambridge University Press, 2007.
  • Buehler, R. J.: Confidence intervals for the product of two binomial parameters., Journal of the American Statistical Association, 52 (280): 482–493, 1957.
  • Chung, K. L.: On a stochastic approximation method., The Annals of Mathematical Statistics, 25 (3): 463–483, 1954.
  • Clopper, C. J. and Pearson, E. S.: The use of confidence or fiducial limits illustrated in the case of the binomial., Biometrika, 26 (4): 404–413, 1934.
  • Cox, D. R. and Snell, E. J.:, Analysis of binary data, volume 32. CRC Press, 1989.
  • Davison, A. C. and Hinkley, D. V.:, Bootstrap methods and their application. Cambridge University Press, 1997.
  • Fabian, V.: On asymptotic normality in stochastic approximation., The Annals of Mathematical Statistics, 39 (4): 1327–1332, 1968.
  • Garthwaite, P. H. and Buckland, S. T.: Generating Monte Carlo confidence intervals by the Robbins-Monro process., Journal of the Royal Statistical Society: Series C (Applied Statistics), 41 (1): 159–171, 1992.
  • Garthwaite, P. H. and Jones, M. C.: A stochastic approximation method and its application to confidence intervals., Journal of Computational and Graphical Statistics, 18 (1): 184–200, 2009.
  • Gordon, T. and Foss, B. M.: The role of stimulation in the delay of onset of crying in new-born infants., J. Exp. Psychol., 16: 79–81, 1966.
  • Jourdain, B. and Lelong, J.: Robust adaptive importance sampling for normal random vectors., The Annals of Applied Probability, 19 (5): 1687–1718, 2009.
  • Kabaila, P. and Lloyd, C. J.: The importance of the designated statistic on Buehler upper limits on a system failure probability., Technometrics, 44 (4), 2002.
  • Kabaila, P. V. and Lloyd, C. J.: Profile upper confidence limits from discrete data., Austral.NZ J. Statist., 42: 67–80, 2001.
  • Kallianpur, G.: A note on the Robbins-Monro stochastic approximation method., The Annals of Mathematical Statistics, 25 (2): 386–388, 1954.
  • Kiefer, J. and Wolfowitz, J.: Stochastic estimation of the maximum of a regression function., The Annals of Mathematical Statistics, 23 (3): 462–466, 1952.
  • Kroese, D. P., Taimre, T., and Botev, Z. I.:, Handbook of Monte Carlo Methods, volume 706. John Wiley & Sons, 2011.
  • Kushner, H. J. and Yang, J.: Stochastic approximation with averaging of the iterates: Optimal asymptotic rate of convergence for general processes., SIAM Journal on Control and Optimization, 31 (4): 1045–1062, 1993.
  • Lemaire, V. and Pages, G.: Unconstrained recursive importance sampling., The Annals of Applied Probability, 20 (3): 1029–1067, 2010.
  • O’Gorman, T. W.: Regaining confidence in confidence intervals for the mean treatment effect., Statistics in medicine, 33 (22): 3859–3868, 2014.
  • Polyak, B. T. and Juditsky, A. B.: Acceleration of stochastic approximation by averaging., SIAM Journal on Control and Optimization, 30 (4): 838–855, 1992.
  • Robbins, H. and Monro, S.: A stochastic approximation method., The Annals of Mathematical statistics, 22 (3): 400–407, 1951.
  • Venter, J. H.: An extension of the robbins-monro procedure., The Annals of Mathematical Statistics, pages 181–190, 1967.
  • Wetherill, G. B.:, Sequential methods in statistics. Chapman and Hall, second edition, 1975.