A new approach to optimal designs for correlated observations

TL;DR

This paper introduces a continuous-time based method for constructing optimal designs in regression models with correlated errors, offering practical and efficient solutions that outperform traditional numerical approaches.

Contribution

It proposes a novel continuous-time framework for optimal design construction in correlated error models, simplifying implementation and improving efficiency.

Findings

New continuous-time approach yields highly efficient estimators.

Method outperforms traditional numerical optimization in practice.

Numerical examples demonstrate practical advantages.

Abstract

This paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes. In contrast to most of the work in this field, which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity, our approach is based on a continuous time model. We use results from stochastic anal- ysis to identify the best linear unbiased estimator (BLUE) in this model. Based on the BLUE, we construct an efficient linear estimator and corresponding optimal designs in the model for finite sample size by minimizing the mean squared error between the opti- mal solution in the continuous time model and its discrete approximation with respect to the weights (of the linear estimator) and the optimal design points, in particular in the multi-parameter case. In contrast to previous work on the subject the resulting estimators and corresponding optimal designs are very efficient and easy to implement. This means that they are practi- cally not distinguishable from the weighted least squares estimator and the corresponding optimal designs, which have to be found numerically by non-convex discrete optimization. The advantages of the new approach are illustrated in several numerical examples.