Convergence of least squares estimators in the adaptive Wynn algorithm for a class of nonlinear regression models

TL;DR

This paper proves the strong consistency and asymptotic normality of adaptive least squares estimators in nonlinear regression models using the Wynn algorithm, under conditions of saturated identifiability and regularity assumptions.

Contribution

It extends previous work by establishing the convergence properties of estimators under the saturated identifiability condition in nonlinear regression.

Findings

Strong consistency of estimators under saturated identifiability.

Asymptotic normality of estimators with interior true parameters.

Applicability under natural continuity and regularity assumptions.

Abstract

The paper continues the authors' work on the adaptive Wynn algorithm in a nonlinear regression model. In the present paper it is shown that if the mean response function satisfies a condition of `saturated identifiability', which was introduced by Pronzato \cite{Pronzato}, then the adaptive least squares estimators are strongly consistent. The condition states that the regression parameter is identifiable under any saturated design, i.e., the values of the mean response function at any $p$ distinct design points determine the parameter point uniquely where, typically, $p$ is the dimension of the regression parameter vector. Further essential assumptions are compactness of the experimental region and of the parameter space together with some natural continuity assumptions. If the true parameter point is an interior point of the parameter space then under some smoothness assumptions and asymptotic homoscedasticity of random errors the asymptotic normality of adaptive least squares estimators is obtained.