Statistical Papers

, Volume 60, Issue 2, pp 165–177 | Cite as

Locally D-optimal designs for a wider class of non-linear models on the k-dimensional ball

with applications to logit and probit models
  • Martin RadloffEmail author
  • Rainer Schwabe
Regular Article


In this paper we extend the results of Radloff and Schwabe (arXiv:1806.00275, 2018), which could be applied for example to Poisson regression, negative binomial regression and proportional hazard models with censoring, to a wider class of non-linear multiple regression models. This includes the binary response models with logit and probit link besides others. For this class of models we derive (locally) D-optimal designs when the design region is a k-dimensional ball. For the corresponding construction we make use of the concept of invariance and equivariance in the context of optimal designs as in our previous paper. In contrast to the former results the designs will not necessarily be exact designs in all cases. Instead approximate designs can appear. These results can be generalized to arbitrary ellipsoidal design regions.


Binary response models D-optimality k-dimensional ball Logit and probit model Multiple regression models 

Mathematics Subject Classification

62K05 62J12 



  1. Atkinson AC, Fedorov VV, Herzberg AM, Zhang R (2014) Elemental information matrices and optimal experimental design for generalized regression models. J Stat Plan Inference 144:81–91MathSciNetCrossRefzbMATHGoogle Scholar
  2. Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with restricted design spaces. J Am Stat Assoc 101(474):747–759MathSciNetCrossRefzbMATHGoogle Scholar
  3. Dette H, Melas VB, Pepelyshev A (2005) Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors. Ann Stat 33:2758–2788MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with Zernike polynomials. Statistics 41:453–470MathSciNetCrossRefzbMATHGoogle Scholar
  5. Farrell RH, Kiefer J, Walbran A (1967) Optimum multivariate designs. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol 1: statistics. University of California Press, Berkeley, pp 113–138Google Scholar
  6. Fedorov VV (1972) Theory of optimal experiments. Academic Press, New YorkGoogle Scholar
  7. Ford I, Torsney B, Wu C (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems. J R Stat Soc Ser B (Stat Methodol) 54(2):569–583zbMATHGoogle Scholar
  8. Hirao M, Sawa M, Jimbo M (2015) Constructions of \(\phi _p\)-optimal rotatable designs on the ball. Sankhya A 77(1):211–236MathSciNetCrossRefzbMATHGoogle Scholar
  9. Kiefer JC (1961) Optimum experimental designs v, with applications to systematic and rotatable designs. In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol 1. University of California Press, Berkeley, pp 381–405Google Scholar
  10. Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter nonlinear models with application to survival models. Stat Sin 24(1):415–428MathSciNetzbMATHGoogle Scholar
  11. Lau TS (1988) \(d\)-optimal designs on the unit \(q\)-ball. J Stat Plan Inference 19(3):299–315MathSciNetCrossRefzbMATHGoogle Scholar
  12. Pukelsheim F (1993) Optimal design of experiments. Wiley series in probability and statistics. Wiley, New YorkGoogle Scholar
  13. Radloff M, Schwabe R (2016) Invariance and equivariance in experimental design for nonlinear models. In: Kunert J, Müller CH, Atkinson AC (eds) mODa 11—advances in model-oriented design and analysis. Springer, Basel, pp 217–224Google Scholar
  14. Radloff M, Schwabe R (2018) Locally \(d\)-optimal designs for non-linear models on the \(k\)-dimensional ball. arXiv:1806.00275
  15. Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information driven by the linear predictor. Stat Sin 27(3):1371–1384MathSciNetzbMATHGoogle Scholar
  16. Silvey SD (1980) Optimal design: an introduction to the theory for parameter estimation. Chapman and Hall, Boca RatonCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute for Mathematical StochasticsOtto-von-Guericke-UniversityMagdeburgGermany

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