Skip to main content
SpringerLink
Log in

Properties of optimal regression designs under the second-order least squares estimator

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

We investigate properties of optimal designs under the second-order least squares estimator (SLSE) for linear and nonlinear regression models. First we derive equivalence theorems for optimal designs under the SLSE. We then obtain the number of support points in A-, c- and D-optimal designs analytically for several models. Using a generalized scale invariance concept we also study the scale invariance property of D-optimal designs. In addition, numerical algorithms are discussed for finding optimal designs. The results are quite general and can be applied for various linear and nonlinear models. Several applications are presented, including results for fractional polynomial, spline regression and trigonometric regression models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Bose M, Mukerjee R (2015) Optimal design measures under asymmetric errors, with application to binary design points. J Stat Plan Inference 159:28–36

    Article  MathSciNet  Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, New York

    Book  Google Scholar 

  • Dette H, Melas VB, Pepelyshev A (2008) Optimal designs for free knot least squares splines. Stat Sin 18:1047–1062

    MathSciNet  MATH  Google Scholar 

  • Dette H, Melas VB, Wong WK (2005) Optimal designs for goodness of fit of the Michaelis-Menten enzyme kinetic function. J Am Stat Assoc 100:1370–1381

    Article  MathSciNet  Google Scholar 

  • Gao LL, Zhou J (2014) New optimal design criteria for regression models with asymmetric errors. J Stat Plan Inference 149:140–151

    Article  MathSciNet  Google Scholar 

  • Gao LL, Zhou J (2017) D-optimal designs based on the second-order least squares estimator. Stat Pap 58:77–94

    Article  MathSciNet  Google Scholar 

  • Grant MC, Boyd SP (2013) The CVX Users Guide. Release 2.0 (beta), CVX Research, Inc. http://cvxr.com/cvx/doc/CVX.pdf. Accessed 14 Oct 2013

  • Kiefer J (1974) General equivalence theorem for optimum designs (approximate theory). Ann Stat 2:849–879

    Article  Google Scholar 

  • Kiefer J, Wolfowitz J (1959) Optimum designs in regression problems. Ann Math Stat 30:271–294

    Article  MathSciNet  Google Scholar 

  • Mandal S, Torsney B, Chowdhury M (2017) Optimal designs for minimising covariances among parameter estimators in a linear model. Aust N Z J Stat 59:255–273

    Article  MathSciNet  Google Scholar 

  • Paquet-Durand O, Zettel V, Hitzmann B (2015) Optimal experimental design for parameter estimation of the Peleg model. Chemom Intell Lab Syst 140:36–42

    Article  Google Scholar 

  • Royston P, Ambler G, Sauerbrei W (1999) The use of fractional polynomials to model continuous risk variables in epidemiology. Int J Epidemiol 28:964–974

    Article  Google Scholar 

  • Silvey SD (1980) Optimal designs: an introduction to the theory for parameter estimation. Chapman and Hall, London

    Book  Google Scholar 

  • Sturm JF (1999) Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optim Methods Softw 11:625–653

    Article  MathSciNet  Google Scholar 

  • Torsney B, Alahmadi AM (1995) Designing for minimally dependent observations. Stat Sin 5:499–514

    MathSciNet  MATH  Google Scholar 

  • Wang L, Leblanc A (2008) Second-order nonlinear least squares estimation. Ann Inst Stat Math 60:883–900

    Article  MathSciNet  Google Scholar 

  • Ye JJ, Zhou J, Zhou W (2017) Computing A-optimal and E-optimal designs for regression models via semidefinite programming. Commun Stat Simul Comput 46:2011–2024

    Article  MathSciNet  Google Scholar 

  • Yin Y, Zhou J (2017) Optimal designs for regression models using the second-order least squares estimator. Stat Sin 27:1841–1856

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research work is supported by Discovery Grants from the Natural Science and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada

    Chi-Kuang Yeh & Julie Zhou

Authors
  1. Chi-Kuang Yeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julie Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julie Zhou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

362_2018_1076_MOESM1_ESM.pdf

We have provided the MATLAB codes of Example 2 for computing A-, c- and D-optimal designs in the supplementary material. These codes can be modified for finding optimal designs for other models and design spaces.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yeh, CK., Zhou, J. Properties of optimal regression designs under the second-order least squares estimator. Stat Papers 62, 75–92 (2021). https://doi.org/10.1007/s00362-018-01076-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-018-01076-6

Keywords

Mathematics Subject Classification

Search

Navigation