## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 4 (2017), 1579-1608.

### A new approach to optimal designs for correlated observations

Holger Dette, Maria Konstantinou, and Anatoly Zhigljavsky

#### 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 analysis 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 optimal 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 multiparameter 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 practically not distinguishable from the weighted least squares estimator and the corresponding optimal designs, which have to be found numerically by nonconvex discrete optimization. The advantages of the new approach are illustrated in several numerical examples.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 4 (2017), 1579-1608.

**Dates**

Received: November 2015

Revised: June 2016

First available in Project Euclid: 28 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1498636867

**Digital Object Identifier**

doi:10.1214/16-AOS1500

**Mathematical Reviews number (MathSciNet)**

MR3670189

**Zentralblatt MATH identifier**

06773284

**Subjects**

Primary: 62K05: Optimal designs

Secondary: 62M05: Markov processes: estimation

**Keywords**

Linear regression correlated observations optimal design Gaussian white mouse model Doob representation quadrature formulas

#### Citation

Dette, Holger; Konstantinou, Maria; Zhigljavsky, Anatoly. A new approach to optimal designs for correlated observations. Ann. Statist. 45 (2017), no. 4, 1579--1608. doi:10.1214/16-AOS1500. https://projecteuclid.org/euclid.aos/1498636867