PICS: A sequential approach to obtain optimal designs for non-linear models leveraging closed-form solutions for faster convergence

TL;DR

This paper introduces PICS, a hybrid sequential method for designing non-linear models that leverages closed-form solutions and current estimates to improve efficiency and reduce computation time.

Contribution

The paper proposes a novel hybrid sequential strategy called PICS that uses closed-form solutions with current parameter estimates for faster, more efficient D-optimal design in non-linear models.

Findings

PICS reduces computational time compared to standard methods.

PICS achieves greater estimation efficiency.

Simulations demonstrate the effectiveness of PICS across models.

Abstract

D-Optimal designs for estimating parameters of response models are derived by maximizing the determinant of the Fisher information matrix. For non-linear models, the Fisher information matrix depends on the unknown parameter vector of interest, leading to a weird situation that in order to obtain the D-optimal design, one needs to have knowledge of the parameter to be estimated. One solution to this problem is to choose the design points sequentially, optimizing the D-optimality criterion using parameter estimates based on available data, followed by updating the parameter estimates using maximum likelihood estimation. On the other hand, there are many non-linear models for which closed-form results for D-optimal designs are available, but because such solutions involve the parameters to be estimated, they can only be used by substituting "guestimates" of parameters. In this paper, a hybrid sequential strategy called PICS (Plug into closed-form solution) is proposed that replaces the optimization of the objective function at every single step by a draw from the probability distribution induced by the known optimal design by plugging in the current estimates. Under regularity conditions, asymptotic normality of the sequence of estimators generated by this approach are established. Usefulness of this approach in terms of saving computational time and achieving greater efficiency of estimation compared to the standard sequential approach are demonstrated with simulations conducted from two different sets of models.