Optimum chemical balance weighing designs with diagonal variance-covariance matrix of errors

Bronisław Ceranka; Małgorzata Graczyk

Discussiones Mathematicae Probability and Statistics (2004)

  • Volume: 24, Issue: 2, page 215-232
  • ISSN: 1509-9423

Abstract

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In this paper we study the estimation problem of individual measurements (weights) of objects in a model of chemical balance weighing design with diagonal variance - covariance matrix of errors under the restriction k₁ + k₂ < p, where k₁ and k₂ represent the number of objects placed on the right and left pans, respectively. We want all variances of estimated measurments to be equal and attaining their lower bound. We give a necessary and sufficient condition under which this lower bound is attained by the variance of each of the estimated measurements. To construct the design matrix X of the considered optimum chemical balance weighing design we use the incidence matrices of balanced bipartite weighing designs.

How to cite

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Bronisław Ceranka, and Małgorzata Graczyk. "Optimum chemical balance weighing designs with diagonal variance-covariance matrix of errors." Discussiones Mathematicae Probability and Statistics 24.2 (2004): 215-232. <http://eudml.org/doc/287753>.

@article{BronisławCeranka2004,
abstract = {In this paper we study the estimation problem of individual measurements (weights) of objects in a model of chemical balance weighing design with diagonal variance - covariance matrix of errors under the restriction k₁ + k₂ < p, where k₁ and k₂ represent the number of objects placed on the right and left pans, respectively. We want all variances of estimated measurments to be equal and attaining their lower bound. We give a necessary and sufficient condition under which this lower bound is attained by the variance of each of the estimated measurements. To construct the design matrix X of the considered optimum chemical balance weighing design we use the incidence matrices of balanced bipartite weighing designs.},
author = {Bronisław Ceranka, Małgorzata Graczyk},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {balanced bipartite weighing design; chemical balanceweighing design; balanced bipartite weighing designs},
language = {eng},
number = {2},
pages = {215-232},
title = {Optimum chemical balance weighing designs with diagonal variance-covariance matrix of errors},
url = {http://eudml.org/doc/287753},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Bronisław Ceranka
AU - Małgorzata Graczyk
TI - Optimum chemical balance weighing designs with diagonal variance-covariance matrix of errors
JO - Discussiones Mathematicae Probability and Statistics
PY - 2004
VL - 24
IS - 2
SP - 215
EP - 232
AB - In this paper we study the estimation problem of individual measurements (weights) of objects in a model of chemical balance weighing design with diagonal variance - covariance matrix of errors under the restriction k₁ + k₂ < p, where k₁ and k₂ represent the number of objects placed on the right and left pans, respectively. We want all variances of estimated measurments to be equal and attaining their lower bound. We give a necessary and sufficient condition under which this lower bound is attained by the variance of each of the estimated measurements. To construct the design matrix X of the considered optimum chemical balance weighing design we use the incidence matrices of balanced bipartite weighing designs.
LA - eng
KW - balanced bipartite weighing design; chemical balanceweighing design; balanced bipartite weighing designs
UR - http://eudml.org/doc/287753
ER -

References

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  1. [1] K.S. Banerjee, Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics. Marcel Dekker Inc., New York 1975. Zbl0334.62030
  2. [2] B. Ceranka and M. Graczyk, Optimum chemical balance weighing designs under the restriction on weighings, Discussiones Mathematicae - Probability and Statistics 21 (2001), 111-120. Zbl1016.05009
  3. [3] B. Ceranka and K. Katulska, Chemical balance weighing designs under the restriction on the number of objects placed on the pans, Tatra Mt. Math. Publ. 17 (1999), 141-148. Zbl0988.62047
  4. [4] B. Ceranka, K. Katulska and D. Mizera, The application of ternary balanced block designs to chemical balance weighing designs, Discussiones Mathematicae - Algebra and Stochastic Methods 18 (1998), 179-185. Zbl0922.62074
  5. [5] H. Hotelling, Some improvements in weighing and other experimental techniques, Ann. Math. Stat. 15 (1944), 297-305. Zbl0063.02076
  6. [6] C. Huang, Balanced bipartite weighing designs, Journal of Combinatorial Theory (A) 21 (1976), 20-34. Zbl0335.05017
  7. [7] K. Katulska, Optimum chemical balance weighing designs with non - homegeneity of the variances of errors, J. Japan Statist. Soc. 19 (1989), 95-101. Zbl0715.62147
  8. [8] J.W. Linnik, Metoda Najmniejszych Kwadratów i Teoria Opracowywania Obserwacji, PWN, Warszawa 1962. 
  9. [9] D. Raghavarao, Constructions and Combinatorial Problems in Design ofExperiments, John Wiley Inc., New York 1971. 
  10. [10] C.R. Rao, Linear Statistical Inference and its Applications, Second Edition, John Wiley and Sons, Inc., New York 1973. Zbl0256.62002
  11. [11] M.N. Swamy, Use of balanced bipartite weighing designs as chemical balance designs, Comm. Statist. Theory Methods 11 (1982), 769-785. Zbl0514.62086

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