1 A stochastic approximation method and its application to confidence intervals By PAUL H. GARTHWAITE and M. C. JONES Department of Statistics, The Open University, Milton Keynes, MK7 6AA, U.K. p.h.garthwaite@open.ac.uk and m.c.jones@open.ac.uk Summary 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 mod- erately long searches. The results motivate the development of a new method of searching for the endpoints of a confidence interval. This method performs de- cidedly 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%. Some key words: Confidence interval; Monte Carlo method; Robbins-Monro; Stochastic approximation. 1. Introduction Stochastic approximation was introduced by Robbins and Monro (1951) for the following problem. Let A(θ) be a fixed but unknown monotonically increasing