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New lower bounds of four-level and two-level designs via two transformations

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Abstract

Code theory is widely used to construct optimal designs in recent years. In this paper, two transformations, a modified Gray map and a mapping between quaternary codes and the sequence of three binary codes for four-level designs, are considered. Via the two transformations, we point out that the wrap-around \(L_2\)-discrepancy values of the two-level designs corresponding to a four-level design are decided by the four-level design, two new analytical expressions of the wrap-around \(L_2\)-discrepancy for the derived two-level designs are built, and some new lower bounds of the wrap-around \(L_2\)-discrepancy for four-level and two-level designs are obtained, which can be used as a benchmark for search the uniform designs and evaluate the uniformity of designs. Furthermore, based on the second transformation, we provide a very strong link between the aberration of a four-level design and the uniformity of the derived two-level design.

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References

  • Chatterjee K, Li ZH, Qin H (2012) Some new lower bounds to centered and wrap-round \(L_2\)-discrepancies. Stat Prob Lett 82:1367–1373

    Article  Google Scholar 

  • Chatterjee K, Ou ZJ, Phoa FKH, Qin H (2017) Uniform four-level designs from two-level designs: a new look. Stat Sin 27:171–186

    MathSciNet  MATH  Google Scholar 

  • Elsawah AM, Qin H (2014) New lower bound for centered \(L_2\)-discrepancy of four-level U-type designs. Stat Prob Lett 93:65–71

    Article  Google Scholar 

  • Fang KT (1980) The uniform design: application of number-theoretic methods in experimental design. Acta Math Appl Sin 3:363–372

    Google Scholar 

  • Fang KT, Li R, Sudjianto A (2005a) Design and modeling for computer experiments. Chapman and Hall, London

    Book  Google Scholar 

  • Fang KT, Tang Y, Yin JX (2005b) Lower bounds for wrap-around \(L_2\)-discrepancy and constructions of symmetrical uniform designs. J Complex 21:757–771

    Article  Google Scholar 

  • Fang KT, Lu X, Winker P (2003) Lower bounds for centered and wrap-around \(L_2\)-discrepancies and construction of uniform design by threshold accepting. J Complex 19:692–711

    Article  Google Scholar 

  • Hickernell FJ (1998a) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322

    Article  MathSciNet  Google Scholar 

  • Hickernell FJ (1998b) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets. Springer, New York, pp 109–166

    Chapter  Google Scholar 

  • Mukerjee R, Wu CFJ (2006) A modern theory of factorial designs. Springer, New York

    MATH  Google Scholar 

  • Phoa FKH (2012) A code arithmetic approach for quaternary code designs and its application to \((1/64)\)th-fraction. Ann Stat 40:3161–3175

    Article  MathSciNet  Google Scholar 

  • Phoa FKH, Xu H (2009) Quarter-fraction factorial designs constructed via quaternary codes. Ann Stat 37:2561–2581

    Article  MathSciNet  Google Scholar 

  • Tang Y, Xu H, Lin DKJ (2012) Uniform fractional factorial designs. Ann Stat 40:891–907

    Article  MathSciNet  Google Scholar 

  • Xu H, Wu CFJ (2001) Generalized minimum aberration for asymmetrical fractional factorial designs. Ann Stat 29:1066–1077

    Article  MathSciNet  Google Scholar 

  • Xu H, Wong A (2007) Two-level nonregular designs from quaternary linear codes. Stat Sin 17:1191–1213

    MathSciNet  MATH  Google Scholar 

  • Xu H, Phoa FKH, Wong WK (2009) Recent developments in nonregular fractional factorial designs. Stat Surv 3:18–46

    Article  MathSciNet  Google Scholar 

  • Zhang RC, Phoa FKH, Mukerjee R, Xu H (2011) A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions. Ann Stat 39:931–955

    Article  MathSciNet  Google Scholar 

  • Zhou YD, Ning JH (2008) Lower bounds of the wrap-around \(L_2\)-discrepancy and relationships between MLHD and uniform design with a large size. J Stat Plan Inference 138:2330–2339

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank an associate editor and the referees for their comments and suggestions which helped to improve this paper. This work was partially supported by the National Natural Science Foundation of China (Nos. 11701213; 11561025), Natural Science Foundation of Hunan Province (Nos. 2017JJ2218; 2017JJ3253) and Excellent Doctor Degree Dissertation Training Project of Central China Normal University (No. 2017YBZZ060).

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Authors and Affiliations

  1. Normal College, Jishou University, Jishou, 416000, China

    Hongyi Li

  2. College of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Hongyi Li & Hong Qin

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  1. Hongyi Li
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  2. Hong Qin
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Correspondence to Hong Qin.

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Li, H., Qin, H. New lower bounds of four-level and two-level designs via two transformations. Stat Papers 61, 1231–1243 (2020). https://doi.org/10.1007/s00362-018-0987-z

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  • DOI: https://doi.org/10.1007/s00362-018-0987-z

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