Abstract
In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms \({\varepsilon_{i},\ i=1,2,\dots,n,}\) are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix \({\widetilde{\bf X}}\) and we prove that this design is D-optimal in certain classes of designs for \({\rho\in[0,1)}\) and it is also D-optimal in the class of designs with the design matrix \({{\bf X} \in M_{n\times 3}(\pm 1)}\) for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs \({M_{n\times 3}(\pm 1)}\) , if \({\rho\in[0,1/(n-2))}\) . We present also the matrix of the D-optimal factorial design with 3 two-level factors.
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Authors are thankful to the referees for their constructive suggestions which improved the presentation of this article.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Katulska, K., Smaga, Ł. D-optimal chemical balance weighing designs with autoregressive errors. Metrika 76, 393–407 (2013). https://doi.org/10.1007/s00184-012-0394-8
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DOI: https://doi.org/10.1007/s00184-012-0394-8