A stochastic approximation method and its application to confidence intervals

Garthwaite, Paul H. and Jones, M. C. (2009). A stochastic approximation method and its application to confidence intervals. Journal of Computational and Graphical Statistics, 18(1) pp. 184–200.

DOI: https://doi.org/10.1198/jcgs.2009.0011


The oldest stochastic approximation method is the Robbins–Monro process. This estimates an unknown scalar parameter by stepping from one trial value for the parameter to another, adopting the last trial value as the estimate. More recent research suggests there are benefits from taking larger steps than with the Robbins–Monro process and then obtaining an estimate by averaging the later trial values. Work on the averaged estimator has made only general assumptions and here we consider a more explicit case that is of practical importance. Stronger asymptotic results are developed and simulations show they hold well for moderately long searches. The results motivate the development of a new method of searching for the endpoints of a confidence interval. This method performs decidedly better than a previously proposed method in terms of both the position of endpoints and the coverage of confidence intervals. The efficiency of the new method is typically well in excess of 90%.

Computer code and an appendix containing the derivation of asymptotic efficiencies are available in the online supplements.

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