Robust prediction and extrapolation designs for nonlinear regression with imprecision
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Robust prediction and extrapolation designs for nonlinear regression with imprecision Xiaojian Xu · Arnold Chen
Received: 22 July 2013 / Accepted: 1 October 2013 © Sapienza Università di Roma 2013
Abstract We consider the general situation of fitting an assumed nonlinear regression model which is possibly misspecified. The minimax designs for both response prediction and extrapolation in biased nonlinear regression models are discussed. We extend previous work of others from linear response or a given function of linear response to intrinsically nonlinear response. Several examples are illustrated such as designing for a yield-fertilizer model, a simple compartmental model, and a Michaelis–Menten model. Keywords Regression design · Nonlinear least squares · Heteroscedasticity · Nonsmooth optimization Mathematics Subject Classification (2000) Secondary 62J12
Primary 62K05 · 62F35;
1 Introduction In this paper, we discuss the construction of robust designs for both prediction and extrapolation of general nonlinear regression responses. The response fitted by the experimenter is a prespecified nonlinear function of unknown parameters and known regressors. Our resulting designs obtained from the proposed construction approaches take into account, and are therefore robust against, both possible imprecision in the specification of the nonlinear regression function, and possible heteroscedasticity.
X. Xu (B) Department of Mathematics, Brock University, 500 Glenridge Ave., St. Catharines, ON L2S 3A1, Canada e-mail: [email protected] A. Chen Department of Mathematics, Brock University, St. Catharines, Canada e-mail: [email protected]
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X. Xu, A. Chen
There are sufficient articles addressing the robust designs against possible misspecified parameter values upon which an optimal design for a nonlinear regression typically depends. In the literature, three ways have been discussed to address such dependency problems. They are a minimax (or maximin) approach (for example [5]), a Bayesian approach (for example [12]), and a sequential (adaptive) design approach (for instance [3]). There are also some but fewer papers addressing the design’s robustness against possible imprecision in the assumed nonlinear regression function. Sinha and Wiens [15] investigate the robust sequential designs for nonlinear regression when the regression function is possibly misspecified. Wiens and Xu [21] discuss the construction of robust designs for a possibly misspecified nonlinear function with a restriction. They have also reviewed robust designs for a possibly misspecified linear response, optimal static and sequential designs for nonlinear regression without the consideration of model uncertainty, and robust sequential designs for nonlinear regressions with such consideration. Please also see the references cited in. Most recently, Karami and Wiens [11] discuss the robust static designs for possibly misspecified nonlinear regression models with a combined Bayesian-minimax procedure. In this paper, we extend the approach of [21] b
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