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Sharp lower bounds of various uniformity criteria for constructing uniform designs

Author

Listed:
  • A. M. Elsawah

    (BNU-HKBU United International College
    Zagazig University)

  • Kai-Tai Fang

    (BNU-HKBU United International College
    The Chinese Academy of Sciences)

  • Ping He

    (BNU-HKBU United International College)

  • Hong Qin

    (Central China Normal University
    Zhongnan University of Economics and Law)

Abstract

Several techniques are proposed for designing experiments in scientific and industrial areas in order to gain much effective information using a relatively small number of trials. Uniform design (UD) plays a significant role due to its flexibility, cost-efficiency and robustness when the underlying models are unknown. UD seeks its design points to be uniformly scattered on the experimental domain by minimizing the deviation between the empirical and theoretical uniform distribution, which is an NP hard problem. Several approaches are adopted to reduce the computational complexity of searching for UDs. Finding sharp lower bounds of this deviation (discrepancy) is one of the most powerful and significant approaches. UDs that involve factors with two levels, three levels, four levels or a mixture of these levels are widely used in practice. This paper gives new sharp lower bounds of the most widely used discrepancies, Lee, wrap-around, centered and mixture discrepancies, for these types of designs. Necessary conditions for the existence of the new lower bounds are presented. Many results in recent literature are given as special cases of this study. A critical comparison study between our results and the existing literature is provided. A new effective version of the fast local search heuristic threshold accepting can be implemented using these new lower bounds. Supplementary material for this article is available online.

Suggested Citation

  • A. M. Elsawah & Kai-Tai Fang & Ping He & Hong Qin, 2021. "Sharp lower bounds of various uniformity criteria for constructing uniform designs," Statistical Papers, Springer, vol. 62(3), pages 1461-1482, June.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:3:d:10.1007_s00362-019-01143-6
    DOI: 10.1007/s00362-019-01143-6
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    References listed on IDEAS

    as
    1. A. M. Elsawah & Kai-Tai Fang, 2018. "New results on quaternary codes and their Gray map images for constructing uniform designs," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(3), pages 307-336, April.
    2. E. Androulakis & K. Drosou & C. Koukouvinos & Y.-D. Zhou, 2016. "Measures of uniformity in experimental designs: A selective overview," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(13), pages 3782-3806, July.
    3. Elsawah, A.M., 2016. "Constructing optimal asymmetric combined designs via Lee discrepancy," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 24-31.
    4. Fred J. Hickernell, 2002. "Uniform designs limit aliasing," Biometrika, Biometrika Trust, vol. 89(4), pages 893-904, December.
    5. Chatterjee, Kashinath & Li, Zhaohai & Qin, Hong, 2012. "Some new lower bounds to centered and wrap-round L2-discrepancies," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1367-1373.
    6. A. M. Elsawah & Hong Qin, 2017. "Optimum mechanism for breaking the confounding effects of mixed-level designs," Computational Statistics, Springer, vol. 32(2), pages 781-802, June.
    7. A. M. Elsawah & Kai-Tai Fang, 2019. "A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs," Journal of Applied Statistics, Taylor & Francis Journals, vol. 46(7), pages 1288-1322, May.
    8. Zhou, Yong-Dao & Ning, Jian-Hui & Song, Xie-Bing, 2008. "Lee discrepancy and its applications in experimental designs," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1933-1942, September.
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