Optimal designs in regression with correlated errors

TL;DR

This paper develops optimal experimental designs for regression models with correlated errors, providing explicit formulas and demonstrating asymptotic efficiency of new estimators compared to traditional methods.

Contribution

It offers a complete solution for optimal designs in correlated regression models, including explicit formulas and asymptotic efficiency results.

Findings

Explicit expressions for the BLUE in continuous models

Optimal designs that asymptotically match the efficiency of the best unbiased estimators

Numerical examples demonstrating near-optimal finite-sample performance

Abstract

This paper discusses the problem of determining optimal designs for regression models, when the observations are dependent and taken on an interval. A complete solution of this challenging optimal design problem is given for a broad class of regression models and covariance kernels. We propose a class of estimators which are only slightly more complicated than the ordinary least-squares estimators. We then demonstrate that we can design the experiments, such that asymptotically the new estimators achieve the same precision as the best linear unbiased estimator computed for the whole trajectory of the process. As a by-product we derive explicit expressions for the BLUE in the continuous time model and analytic expressions for the optimal designs in a wide class of regression models. We also demonstrate that for a finite number of observations the precision of the proposed procedure, which includes the estimator and design, is very close to the best achievable. The results are illustrated on a few numerical examples.