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Lee discrepancy and its applications in experimental designs

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  • Zhou, Yong-Dao
  • Ning, Jian-Hui
  • Song, Xie-Bing

Abstract

Various discrepancies have been defined in uniform designs, such as centered L2-discrepancy, wrap-around L2-discrepancy and discrete discrepancy. Among them the discrete discrepancy can explore relationships among uniform designs, fractional factorial designs, and combinational designs. However, the discrete discrepancy is mainly good for two-level factorial designs. In this paper, a new discrepancy based on the Lee distance, Lee discrepancy, is proposed and its computational formula is given. The Lee discrepancy can expand the relationships between the discrete discrepancy and some criteria for factorial designs with multiple levels. Some lower bounds of the Lee discrepancy for symmetrical and asymmetrical designs are given, and some connections between the Lee discrepancy and the generalized minimum aberration are considered. Finally, relationships between the Lee discrepancy and majorization framework are also considered.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:13:p:1933-1942
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    References listed on IDEAS

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    1. Kai-Tai Fang & Dennis K. J. Lin & Min-Qian Liu, 2003. "Optimal mixed-level supersaturated design," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 58(3), pages 279-291, December.
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    1. Liuping Hu & Zujun Ou & Hongyi Li, 2020. "Construction of four-level and mixed-level designs with zero Lee discrepancy," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(1), pages 129-139, January.
    2. Elsawah, A.M., 2016. "Constructing optimal asymmetric combined designs via Lee discrepancy," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 24-31.
    3. Biao Luo & Hongyi Li & Yingying Wei & Zujun Ou, 2022. "Uniform design with prior information of factors under weighted wrap-around $$L_2$$ L 2 -discrepancy," Computational Statistics, Springer, vol. 37(5), pages 2717-2739, November.
    4. A. M. Elsawah, 2018. "Choice of optimal second stage designs in two-stage experiments," Computational Statistics, Springer, vol. 33(2), pages 933-965, June.
    5. Yan-Ping Gao & Si-Yu Yi & Yong-Dao Zhou, 2022. "Level-augmented uniform designs," Statistical Papers, Springer, vol. 63(2), pages 441-460, April.
    6. Zou, Na & Ren, Ping & Qin, Hong, 2009. "A note on Lee discrepancy," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 496-500, February.
    7. 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.
    8. Zou, Na & Gou, Tingxun & Qin, Hong & Chatterjee, Kashinath, 2020. "Generalized foldover method for high-level designs," Statistics & Probability Letters, Elsevier, vol. 164(C).
    9. Elsawah, A.M. & Qin, Hong, 2015. "A new strategy for optimal foldover two-level designs," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 116-126.
    10. Elsawah, A.M. & Qin, Hong, 2015. "Lee discrepancy on symmetric three-level combined designs," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 273-280.
    11. Kang Wang & Zujun Ou & Jiaqi Liu & Hongyi Li, 2021. "Uniformity pattern of q-level factorials under mixture discrepancy," Statistical Papers, Springer, vol. 62(4), pages 1777-1793, August.
    12. 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.
    13. Liuping Hu & Hongyi Li & Zujun Ou, 2019. "Constructing optimal four-level designs via Gray map code," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(5), pages 573-587, July.
    14. Yong-Dao Zhou & Hongquan Xu, 2014. "Space-Filling Fractional Factorial Designs," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1134-1144, September.
    15. Liuping Hu & Kashinath Chatterjee & Jiaqi Liu & Zujun Ou, 2020. "New lower bound for Lee discrepancy of asymmetrical factorials," Statistical Papers, Springer, vol. 61(4), pages 1763-1772, August.

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