Optimal designs for regression models with autoregressive errors structure

TL;DR

This paper derives explicit optimal designs for regression models with AR(1) and AR(2) errors, providing nearly optimal estimators and designs with high efficiency in finite samples and continuous models.

Contribution

It introduces explicit formulas and continuous approximations for optimal designs in regression models with autoregressive errors, enhancing estimator efficiency.

Findings

Optimal designs closely match weighted least squares efficiency.

Derived estimators achieve near-optimal variance in finite samples.

Illustrated methods with several practical examples.

Abstract

In the one-parameter regression model with AR(1) and AR(2) errors we find explicit expressions and a continuous approximation of the optimal discrete design for the signed least square estimator. The results are used to derive the optimal variance of the best linear estimator in the continuous time model and to construct efficient estimators and corresponding optimal designs for finite samples. The resulting procedure (estimator and design) provides nearly the same efficiency as the weighted least squares and its variance is close to the optimal variance in the continuous time model. The results are illustrated by several examples demonstrating the feasibility of our approach.