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September, 1952 On the Stochastic Approximation Method of Robbins and Monro
J. Wolfowitz
Ann. Math. Statist. 23(3): 457-461 (September, 1952). DOI: 10.1214/aoms/1177729391

Abstract

In their interesting and pioneering paper Robbins and Monro [1] give a method for "solving stochastically" the equation in $x: M(x) = \alpha$, where $M(x)$ is the (unknown) expected value at level $x$ of the response to a certain experiment. They raise the question whether their results, which are contained in their Theorems 1 and 2, are valid under a condition (their condition (4'), our condition (1) below) which is statistically plausible and is weaker than the condition which they require to prove their results. In the present paper this question is answered in the affirmative. They also ask whether their conditions (33), (34), and (35) (our conditions (25), (26) and (27) below) can be replaced by their condition (5") (our condition (28) below). A counterexample shows that this is impossible. However, it is possible to weaken conditions (25), (26) and (27) by replacing them by condition (3) (abc) below. Thus our results generalize those of [1]. The statistical significance of these results is described in [1].

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J. Wolfowitz. "On the Stochastic Approximation Method of Robbins and Monro." Ann. Math. Statist. 23 (3) 457 - 461, September, 1952. https://doi.org/10.1214/aoms/1177729391

Information

Published: September, 1952
First available in Project Euclid: 28 April 2007

zbMATH: 0049.36505
MathSciNet: MR50242
Digital Object Identifier: 10.1214/aoms/1177729391

Rights: Copyright © 1952 Institute of Mathematical Statistics

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Vol.23 • No. 3 • September, 1952
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