Notes on D‑optimal Spring Balance Weighing Designs

Bronisław Ceranka; Małgorzata Graczyk

doi:10.18778/0208-6018.338.11

Authors

Bronisław Ceranka Poznań University of Life Sciences, Faculty of Agronomy and Bioengineering, Department of Mathematical and Statistical Methods
Małgorzata Graczyk Poznań University of Life Sciences, Faculty of Agronomy and Bioengineering, Department of Mathematical and Statistical Methods

DOI:

https://doi.org/10.18778/0208-6018.338.11

Keywords:

D-optimal design, spring balance weighing design

Abstract

Spring balance weighing design is a model of an experiment in which the result can be presented as a linear combination of unknown measurements of objects with factors of this combination equalling zero or one. In this paper, we assume that the variances of measurement errors are not equal and errors are not correlated. We consider D‑optimal designs, i.e. designs in which the determinant of the information matrix for the design attains the maximal value. The upper bound of its value is obtained and the conditions for the upper bound to be attained are proved. The value of the upper bound depends on whether the number of objects in the experiment is odd or even. Some methods of construction of regular D‑optimal spring balance weighing designs are demonstrated.

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References

Banerjee K.S. (1975), Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics, Marcel Dekker Inc., New York.
Google Scholar

Ceranka B., Graczyk M. (2013), Construction of E‑optimal spring balance weighing designs for even number of objects, “Acta Universitatis Lodziensis. Folia Oeconomica”, vol. 285, pp. 141–148.
Google Scholar

Ceranka B., Graczyk M. (2014), Regular D‑optimal spring balance weighing designs: construction, “Acta Universitatis Lodziensis. Folia Oeconomica”, vol. 302, pp. 111–125.
Google Scholar

Ceranka B., Graczyk M. (2015), On D‑optimal chemical balance weighing designs, “Acta Universitatis Lodziensis. Folia Oeconomica”, vol. 311, pp. 71–84.
Google Scholar

Ceranka B., Graczyk M. (2016), About some properties and constructions of experimental designs, “Acta Universitatis Lodziensis. Folia Oeconomica”, vol. 333, pp. 73–85.
Google Scholar

Ceranka B., Graczyk M. (2017), Recent developments in D–optimal spring balance weighing designs, “Communication in Statistics‑Theory and Methods”, accepted to publication.
Google Scholar

Ceranka B., Graczyk M., Katulska K. (2009), On some constructions of regular D–optimal spring balance weighing designs, “Biometrical Letters”, vol. 46, pp. 103–112.
Google Scholar

Cheng C.S. (2014), Optimal biased weighing designs and two‑level main effect plans, “Journal of Statistical Theory and Practice”, vol. 8, pp. 83–99.
Google Scholar

Harville D.A. (1997), Matrix Algebra from a Statistician’s Perspective, Springer Verlag, New York.
Google Scholar

Hudelson M., Klee V., Larman D. (1996), Largest j‑simplices in d‑cubes: Some relatives to the Hadamard determinant problem, “Linear Algebra and its Applications”, vol. 24, pp. 519–598.
Google Scholar

Jacroux M., Notz W. (1983), On the optimality of spring balance weighing designs, “The Annals of Statistics”, vol. 11, pp. 970–978.
Google Scholar

Katulska K., Przybył K. (2007), On certain D‑optimal spring balance weighing designs, “Journal of Statistical Theory and Practice”, vol. 1, pp. 393–404.
Google Scholar

Masaro J., Wong Ch.S. (2008), D‑optimal designs for correlated random vectors, “Journal of Statistical Planning and Inference”, vol. 138, pp. 4093–4106.
Google Scholar

Neubauer M.G., Watkins W., Zeitlin J. (1997), Maximal j‑simplices in the real dimensional unit cube, “Journal of Combinatorial Theory”, Ser. A, vol. 80, pp. 1–12.
Google Scholar

Neubauer G.N., Watkins W., Zeitlin J. (1998), Notes on D‑optimal designs, “Linear Algebra and its Applications”, vol. 280, pp. 109–127.
Google Scholar

Raghavarao D. (1971), Constructions and combinatorial problems in design of experiment, John Wiley and Sons, New York.
Google Scholar