Journal of Systems Science and Complexity

, Volume 30, Issue 6, pp 1459–1469 | Cite as

Effective lower bounds of wrap-around L2-discrepancy on three-level combined designs

  • A. M. ElsawahEmail author
  • Jianwei Hu
  • Hong Qin


How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions. This paper takes the wrap-around L2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.


Combined design Foldover plan L2-discrepancy lower bound optimal combined design wrap-around 


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  1. [1]
    Hickernell F J, A generalized discrepancy and quadrature error bound, Math. Comp., 1998, 67: 299–322.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Hickernell F J, Lattice rules: How well do they measure up? Random and Quasi-Random Point Sets (eds. by Hellekalek P and Larcher G), Springer, New York, 1998.Google Scholar
  3. [3]
    Box G E P, Hunter W G, and Hunter J S, Statistics for Experiments, John Wiley and Sons, NewYork, 1978.zbMATHGoogle Scholar
  4. [4]
    Montgomery D C and Runger G C, Foldover of 2k−p resolution IV experimental designs, J. Qual. Technol., 1996, 28: 446–450.Google Scholar
  5. [5]
    Li H and Mee R W, Better foldover fractions for resolution III 2k−p designs, Technometrics, 2002, 44: 278–283.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Li W and Lin D K J, Optimal foldover plans for two-Level fractional factorial designs, Technometrics, 2003, 45: 142–149.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Li W, Lin D K J, and Ye K Q, Optimal foldover plans for non-regular orthogonal designs, Technometrics, 2003, 45: 347–351.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Ye K Q and Li W, Some properties of blocked and unblocked foldover of 2k−p designs, Statist. Sinica, 2003, 13: 403–408.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Fang K T, Lin D K J, and Qin H, A note on optimal foldover design, Statist. Probab. Lett., 2003, 62: 245–250.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lei Y J, Qin H, and Zou N, Some lower bounds of centered L2-discrepancy on foldover designs, Acta Math. Sci., 2010, 30A: 1555–1561.Google Scholar
  11. [11]
    Lei Y J, Ou Z J, Qin H, et al., A note on lower bound of centered L2-discrepancy on combined designs, Acta Math. Sin., 2012, 28: 793–800.CrossRefzbMATHGoogle Scholar
  12. [12]
    Ou Z J, Chatterjee K, and Qin H, Lower bounds of various discrepancies on combined designs, Metrika, 2011, 74: 109–119.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Qin H, Chatterjee K, and Ou Z J, A lower bound for the centered L2-discrepancy on combined designs under the asymmetric factorials, Statistics, 2013, 47: 992–1002.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Ou Z J, Qin H, and Cai X, Optimal foldover plans of three level designs with minimum wraparound L2-discrepancy, Sci. China Math., 2015, 58: 1537–1548.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Elsawah A M and Qin H, Lee discrepancy on symmetric three-level combined designs, Statist. Probab. Lett., 2015, 96: 273–280.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Elsawah A M and Qin H, A new strategy for optimal foldover two-level designs, Statist. Probab. Lett., 2015, 103: 116–126.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Elsawah A M and Qin H, An efficient methodology for constructing optimal foldover designs in terms of mixture discrepancy, J. Korean Statist. Soc., 2016, 45: 77–88.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Elsawah A M, Al-awady M A, Abd Elgawad M A, et al., A note on optimal foldover four-level factorials, Acta Math. Sinica, 2016, 32: 286–296.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Elsawah A M and Qin H, Optimum mechanism for breaking the confounding effects of mixed-level designs, Comput. Stat., 2016, doi: 10.1007/s00180–016–0651–9.Google Scholar
  20. [20]
    Fang K T and Hickernell F J, The uniform design and its applications, Bulletin of The International Statistical Institute, 50th Session, 1995, 1: 339–349.Google Scholar
  21. [21]
    Elsawah A M and Qin H, New lower bound for centered L2-discrepancy of four-level U-type designs, Statist. Probab. Lett., 2014, 93: 65–71.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Elsawah A M and Qin H, Lower bound of centered L2-discrepancy for mixed two and three levels U-type designs, J. Statist. Plann. Inference, 2015, 161: 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Elsawah A M and Qin H, Mixture discrepancy on symmetric balanced designs, Statist. Probab. Lett., 2015, 104: 123–132.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Elsawah A M and Qin H, Asymmetric uniform designs based on mixture discrepancy, J. App. Statist., 2016, 43(12): 2280–2294.MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and StatisticsCentral China Normal UniversityWuhanChina
  2. 2.Division of Science and TechnologyBNU-HKBU United International CollegeZhuhaiChina
  3. 3.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt

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