The Annals of Statistics

A new approach to optimal designs for correlated observations

Holger Dette, Maria Konstantinou, and Anatoly Zhigljavsky

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Received: November 2015
Revised: June 2016
First available in Project Euclid: 28 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K05: Optimal designs
Secondary: 62M05: Markov processes: estimation

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


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.

Export citation


  • Akhiezer, N. I. and Glazman, I. M. (1993). Theory of Linear Operators in Hilbert Space. Dover Publications, New York.
  • Bickel, P. J. and Herzberg, A. M. (1979). Robustness of design against autocorrelation in time. I. Asymptotic theory, optimality for location and linear regression. Ann. Statist. 7 77–95.
  • Boltze, L. and Näther, W. (1982). On effective observation methods in regression models with correlated errors. Math. Operationsforsch. Statist. Ser. Statist. 13 507–519.
  • Clerc, M. (2006). Particle Swarm Optimization. ISTE, London.
  • Dette, H., Kunert, J. and Pepelyshev, A. (2008). Exact optimal designs for weighted least squares analysis with correlated errors. Statist. Sinica 18 135–154.
  • Dette, H., Pepelyshev, A. and Zhigljavsky, A. (2013). Optimal design for linear models with correlated observations. Ann. Statist. 41 143–176.
  • Dette, H., Pepelyshev, A. and Zhigljavsky, A. (2016). Optimal designs in regression with correlated errors. Ann. Statist. 44 113–152.
  • Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Stat. 20 393–403.
  • Harman, R. and Štulajter, F. (2010). Optimal prediction designs in finite discrete spectrum linear regression models. Metrika 72 281–294.
  • Harman, R. and Štulajter, F. (2011). Optimality of equidistant sampling designs for the Brownian motion with a quadratic drift. J. Statist. Plann. Inference 141 2750–2758.
  • Ibragimov, I. A. and Has’minskiĭ, R. Z. (1981). Statistical Estimation. Applications of Mathematics 16. Springer, New York.
  • Kiseľák, J. and Stehlík, M. (2008). Equidistant and $D$-optimal designs for parameters of Ornstein–Uhlenbeck process. Statist. Probab. Lett. 78 1388–1396.
  • Mehr, C. B. and McFadden, J. A. (1965). Certain properties of Gaussian processes and their first-passage times. J. R. Stat. Soc. Ser. B. Stat. Methodol. 27 505–522.
  • Müller, W. G. and Pázman, A. (2003). Measures for designs in experiments with correlated errors. Biometrika 90 423–434.
  • Näther, W. (1985a). Effective Observation of Random Fields. BSB B. G. Teubner Verlagsgesellschaft, Leipzig.
  • Näther, W. (1985b). Exact design for regression models with correlated errors. Statistics 16 479–484.
  • Pázman, A. and Müller, W. G. (2001). Optimal design of experiments subject to correlated errors. Statist. Probab. Lett. 52 29–34.
  • Pukelsheim, F. (2006). Optimal Design of Experiments. SIAM, Philadelphia, PA.
  • Sacks, J. and Ylvisaker, D. (1966). Designs for regression problems with correlated errors. Ann. Math. Stat. 37 66–89.
  • Sacks, J. and Ylvisaker, D. (1968). Designs for regression problems with correlated errors; many parameters. Ann. Math. Stat. 39 49–69.
  • Wong, W. K., Chen, R.-B., Huang, C.-C. and Wang, W. (2015). A modified particle swarm optimization technique for finding optimal designs for mixture models. PLoS ONE 10 e0124720.
  • Zhigljavsky, A., Dette, H. R. and Pepelyshev, A. (2010). A new approach to optimal design for linear models with correlated observations. J. Amer. Statist. Assoc. 105 1093–1103.