AI summary

This paper introduces a stochastic approximation method to find the root \(\theta\) of the equation \(M(x) = a\), where \(M(x)\) is a monotone function unknown to the experimenter. The method involves successive experiments at levels \(x_1, x_2, \dots\) such that \(x_n\) converges to \(\theta\) in probability.

Abstract

Let M(x) denote the expected value at level x of the response to a certain experiment. M(x) is assumed to be a monotone function of x but is unknown to the experimenter, and it is desired to find the solution x = 0 of the equation M(x) = a, where a is a given constant. We give a method for making successive experiments at levels X1, X2, in such a way that xn will tend to θ in probability.