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

  • Chi-Kuang Yeh
  • Julie Zhou
Regular Article


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.


A-optimal design Convex optimization D-optimal design Fractional polynomial Generalized scale invariance Peleg model Spline regression Number of support points 

Mathematics Subject Classification

62K05 62K20 



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

Supplementary material

362_2018_1076_MOESM1_ESM.pdf (66 kb)
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.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

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