A Stochastic Approximation by Observations on a Discrete Lattice Using Isotonic Regression

Abstract

A new non-parametric stochastic approximation procedure for estimating the roots of a non-decreasing regression function is described. The observations are taken on a discrete lattice and the estimation is based on the roots of the sample isotonic regression function fitted to the observed values. Asymptotic properties of the estimator are proven. When specialized to the bio-assay case it gives asymptotic results similar to those obtained by Derman for his up-and-down method but under weaker assumptions than Derman required.