A $p$-step-ahead sequential adaptive algorithm for D-optimal nonlinear regression design

TL;DR

This paper introduces a $p$-step-ahead adaptive algorithm for nonlinear regression design that sequentially constructs locally D-optimal designs, ensuring strong consistency and asymptotic optimality under certain conditions.

Contribution

It proposes a novel $p$-step-ahead adaptive design algorithm for nonlinear regression models, extending previous methods to ensure asymptotic D-optimality.

Findings

Adaptive estimators are strongly consistent and asymptotically normal.

The generated design sequence is asymptotically D-optimal under saturated D-optimality.

The algorithm applies to models satisfying saturated identifiability and generalized linear models.

Abstract

Under a nonlinear regression model with univariate response an algorithm for the generation of sequential adaptive designs is studied. At each stage, the current design is augmented by adding $p$ design points where $p$ is the dimension of the parameter of the model. The augmenting $p$ points are such that, at the current parameter estimate, they constitute the locally D-optimal design within the set of all saturated designs. Two relevant subclasses of nonlinear regression models are focused on, which were considered in previous work of the authors on the adaptive Wynn algorithm: firstly, regression models satisfying the `saturated identifiability condition' and, secondly, generalized linear models. Adaptive least squares estimators and adaptive maximum likelihood estimators in the algorithm are shown to be strongly consistent and asymptotically normal, under appropriate assumptions. For both model classes, if a condition of `saturated D-optimality' is satisfied, the almost sure asymptotic D-optimality of the generated design sequence is implied by the strong consistency of the adaptive estimators employed by the algorithm. The condition states that there is a saturated design which is locally D-optimal at the true parameter point (in the class of all designs).