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Local c- and E-optimal Designs for Exponential Regression Models

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Abstract

In this paper we investigate local E- and c-optimal designs for exponential regression models of the form \(\sum_{i=1}^k a_i\exp\left(-\mu_ix\right)\). We establish a numerical method for the construction of efficient and local optimal designs, which is based on two results. First, we consider for fixed k the limit μ i → γ (i = 1, ... , k) and show that the optimal designs converge weakly to the optimal designs in a heteroscedastic polynomial regression model. It is then demonstrated that in this model the optimal designs can be easily determined by standard numerical software. Secondly, it is proved that the support points and weights of the local optimal designs in the exponential regression model are analytic functions of the nonlinear parameters μ 1, ... , μ k . This result is used for the numerical calculation of the local E-optimal designs by means of a Taylor expansion for any vector (μ 1, ... , μ k ). It is also demonstrated that in the models under consideration E-optimal designs are usually more efficient for estimating individual parameters than D-optimal designs.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Holger Dette

  2. Department of Mathematics, St. Petersburg State University, St. Petersburg, Russia

    Viatcheslav B. Melas & Andrey Pepelyshev

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  1. Holger Dette
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  2. Viatcheslav B. Melas
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  3. Andrey Pepelyshev
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Correspondence to Holger Dette.

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Dette, H., Melas, V.B. & Pepelyshev, A. Local c- and E-optimal Designs for Exponential Regression Models. Ann Inst Stat Math 58, 407–426 (2006). https://doi.org/10.1007/s10463-006-0031-2

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  • DOI: https://doi.org/10.1007/s10463-006-0031-2

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