Stochastic Estimation of the Maximum of a Regression Function

Abstract

Let M(x) be a regression function which has a maximum at the unknown point θ. M(x) is itself unknown to the statistician who, however, can take observations at any level x. This paper gives a scheme whereby, starting from an arbitrary point x1, one obtains successively x2, x3, ⋯ such that xn converges to θ in probability as n → ∞.