## The Annals of Statistics

### Optimal designs in regression with correlated errors

#### 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.

#### Article information

Source
Ann. Statist., Volume 44, Number 1 (2016), 113-152.

Dates
Revised: June 2015
First available in Project Euclid: 10 December 2015

https://projecteuclid.org/euclid.aos/1449755959

Digital Object Identifier
doi:10.1214/15-AOS1361

Mathematical Reviews number (MathSciNet)
MR3449764

Zentralblatt MATH identifier
1338.62161

Subjects
Primary: 62K05: Optimal designs
Secondary: 31A10: Integral representations, integral operators, integral equations methods

#### Citation

Dette, Holger; Pepelyshev, Andrey; Zhigljavsky, Anatoly. Optimal designs in regression with correlated errors. Ann. Statist. 44 (2016), no. 1, 113--152. doi:10.1214/15-AOS1361. https://projecteuclid.org/euclid.aos/1449755959

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