zbMATH — the first resource for mathematics

Robust static designs for approximately specified nonlinear regression models. (English) Zbl 1278.62129
Summary: We outline the construction of robust, static designs for nonlinear regression models. The designs are robust in that they afford protection from increases in the mean squared error resulting from misspecifications of the model fitted by the experimenter. This robustness is obtained through a combination of minimax and Bayesian procedures. We first maximize (over a neighborhood of the fitted response function) and then average (with respect to a prior on the parameters) the sum (over the design space) of the mean squared errors of the predictions. This average maximum loss is then minimized over the class of designs. Averaging with respect to a prior means that there is no remaining dependence on unknown parameters, thus allowing for static, rather than sequential, design construction. The minimization over the class of designs is carried out by implementing a genetic algorithm. Several examples are discussed.
Reviewer: Reviewer (Berlin)

62K25 Robust parameter designs
62J02 General nonlinear regression
90C59 Approximation methods and heuristics in mathematical programming
Full Text: DOI
[1] Bates, D. M.; Watts, D. G., Nonlinear regression analysis and its applications, (1988), Wiley · Zbl 0728.62062
[2] Box, G. E.P.; Hunter, W. G., The experimental study of physical mechanisms, Technometrics, 7, 23-42, (1965)
[3] Box, G.E.P., Hunter, W.G., 1965b. Sequential design of experiments for nonlinear models. In: Proceedings of the IBM Scientific Computing Symposium on Statistics, October 21-23, 1963, pp. 113-137.
[4] Dette, H.; Neugebauer, H.-M., Bayesian D-optimal designs for exponential regression models, Journal of Statistical Planning and Inference, 60, 331-349, (1997) · Zbl 0900.62408
[5] Dette, H.; Biedermann, S., Robust and efficient designs for the Michaelis-Menten model, Journal of the American Statistical Association, 98, 679-686, (2003) · Zbl 1040.62065
[6] Dette, H.; Pepelyshev, A., Efficient experimental designs for sigmoidal growth models, Journal of Statistical Planning and Inference, 138, 2-17, (2008) · Zbl 1144.62057
[7] Dette, H., Pepelyshev, A., Zhigljavsky, A., 2008. Improving Updating Rules in Multiplicative Algorithms for Computing D-optimal designs. Computational Statistics and Data Analysis 53, 312-320. · Zbl 1231.62141
[8] Fang, Z.; Wiens, D. P., Integer-valued, minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models, Journal of the American Statistical Association, 95, 807-818, (2000) · Zbl 0995.62068
[9] Gallant, A. R., Nonlinear statistical models, (1987), Wiley · Zbl 0611.62071
[10] Karami, J.H., 2011. Designs for Nonlinear Regression With a Prior on the Parameters. Unpublished MSc Thesis. University of Alberta, Department of Mathematical and Statistical Sciences.
[11] King, J.; Wong, W.-K., Minimax D-optimal designs for the logistic model, Biometrics, 56, 1263-1267, (2000) · Zbl 1060.62545
[12] Li, P.; Wiens, D. P., Robustness of design for dose-response studies, Journal of the Royal Statistical Society (Series B), 17, 215-238, (2011) · Zbl 1411.62323
[13] Mandal, A.; Johnson, K.; Wu, J. C.F.; Bornemeier, D., Identifying promising compounds in drug discoverygenetic algorithms and some new statistical techniques, Journal of Chemical Information and Modeling, 47, 981-988, (2007)
[14] Sinha, S.; Wiens, D. P., Robust sequential designs for nonlinear regression, The Canadian Journal of Statistics, 30, 601-618, (2002) · Zbl 1018.62057
[15] Sinha, S., Wiens, D.P., 2003. Asymptotics for robust sequential designs in misspecified regression models. In: Moore, M., Léger, C., Froda, S. (Eds.), Mathematical Statistics and Applications: Festschrift for Constance van Eeden. IMS Lecture Notes—Monograph Series, pp. 233-248.
[16] Welsh, A.H., Wiens, D.P. Robust model-based sampling designs. Statistics and Computing, 10.1007/s11222-012-9339-3, in press. · Zbl 1322.62064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.