The Annals of Statistics

Optimal designs in regression with correlated errors

Holger Dette, Andrey Pepelyshev, and Anatoly Zhigljavsky

Full-text: Open access

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
Received: January 2015
Revised: June 2015
First available in Project Euclid: 10 December 2015

Permanent link to this document
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

Keywords
Linear regression correlated observations signed measures optimal design BLUE Gaussian processes Doob representation

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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References

  • 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.
  • 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 Holland-Letz, T. (2010). Optimal designs for random effect models with correlated errors with applications in population pharmacokinetics. Ann. Appl. Stat. 4 1430–1450.
  • Dette, H., Pepelyshev, A. and Zhigljavsky, A. (2013). Optimal design for linear models with correlated observations. Ann. Statist. 41 143–176.
  • Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20 393–403.
  • Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195–277.
  • 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.
  • Hughes-Oliver, J. M. (1998). Optimal designs for nonlinear models with correlated errors. In New Developments and Applications in Experimental Design. IMS Lecture Notes Monogr. Ser. 34 163–174. IMS, Hayward, CA.
  • Kiseľák, J. and Stehlík, M. (2008). Equidistant and $D$-optimal designs for parameters of Ornstein–Uhlenbeck process. Statist. Probab. Lett. 78 1388–1396.
  • Lindsey, J. K. (1993). Models for Repeated Measurements. The Clarendon Press, New York.
  • Mehr, C. B. and McFadden, J. A. (1965). Certain properties of Gaussian processes and their first-passage times. J. Roy. Statist. Soc. Ser. B 27 505–522.
  • Mentré, F., Mallet, A. and Baccar, D. (1997). Optimal design in random-effects regression models. Biometrika 84 429–442.
  • Morrison, D. F. (1972). The analysis of a single sample of repeated measurements. Biometrics 28 55–71.
  • 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, N. D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37 66–89.
  • Sacks, J. and Ylvisaker, D. (1968). Designs for regression problems with correlated errors; many parameters. Ann. Math. Statist. 39 49–69.
  • Zhigljavsky, A. A. (1991). Theory of Global Random Search. Kluwer Academic, Dordrecht.
  • Zhigljavsky, A., Dette, H. and Pepelyshev, A. (2010). A new approach to optimal design for linear models with correlated observations. J. Amer. Statist. Assoc. 105 1093–1103.