Abstract
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.
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12 September 2019
Unfortunately, due to a technical error, the articles published in issues 60:2 and 60:3 received incorrect pagination. Please find here the corrected Tables of Contents. We apologize to the authors of the articles and the readers.
References
Atkinson AC, Fedorov VV, Herzberg AM, Zhang R (2014) Elemental information matrices and optimal experimental design for generalized regression models. J Stat Plan Inference 144:81–91
Biedermann S, Dette H, Zhu W (2006) Optimal designs for dose-response models with restricted design spaces. J Am Stat Assoc 101(474):747–759
Dette H, Melas VB, Pepelyshev A (2005) Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors. Ann Stat 33:2758–2788
Dette H, Melas VB, Pepelyshev A (2007) Optimal designs for statistical analysis with Zernike polynomials. Statistics 41:453–470
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 113–138
Fedorov VV (1972) Theory of optimal experiments. Academic Press, New York
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):569–583
Hirao M, Sawa M, Jimbo M (2015) Constructions of \(\phi _p\)-optimal rotatable designs on the ball. Sankhya A 77(1):211–236
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 381–405
Konstantinou M, Biedermann S, Kimber A (2014) Optimal designs for two-parameter nonlinear models with application to survival models. Stat Sin 24(1):415–428
Lau TS (1988) \(d\)-optimal designs on the unit \(q\)-ball. J Stat Plan Inference 19(3):299–315
Pukelsheim F (1993) Optimal design of experiments. Wiley series in probability and statistics. Wiley, New York
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 217–224
Radloff M, Schwabe R (2018) Locally \(d\)-optimal designs for non-linear models on the \(k\)-dimensional ball. arXiv:1806.00275
Schmidt D, Schwabe R (2017) Optimal design for multiple regression with information driven by the linear predictor. Stat Sin 27(3):1371–1384
Silvey SD (1980) Optimal design: an introduction to the theory for parameter estimation. Chapman and Hall, Boca Raton
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Radloff, M., Schwabe, R. Locally D-optimal designs for a wider class of non-linear models on the k-dimensional ball. Stat Papers 60, 515–527 (2019). https://doi.org/10.1007/s00362-018-01078-4
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DOI: https://doi.org/10.1007/s00362-018-01078-4
Keywords
- Binary response models
- D-optimality
- k-dimensional ball
- Logit and probit model
- Multiple regression models