Abstract
Literatures reveal that foldover is a useful technique in construction of factorial designs. The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around \(L_2\)-discrepancy to assess the optimal foldover plans for asymmetric fractional factorials. A general foldover strategy and combined design under a foldover plan are developed for asymmetric fractional factorials, some theoretical properties on the equivalence between the defined foldover plan and its complementary foldover plan are discussed. A new lower bound for the wrap-around \(L_2\)-discrepancy of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.
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References
Box GEP, Hunter WG, Hunter JS (1978) Statistics for experiments. Wiley, New York
Chatterjee K, Fang KT, Qin H (2006) A lower bound for the centered \(L_2\)-discrepancy on asymmetric factorials and its application. Metrika 63:243–255
Fang KT, Lin DKJ, Qin H (2003) A note on optimal foldover design. Statist Probab Lett 62:245–250
Fang KT, Mukerjee R (2000) Connection between uniformity and aberration in regular fractions of two-level factorials. Biometrika 87:173–198
Fang KT, Tang Y, Yin JX (2005) Lower bounds for wrap-around \(L_2\)-discrepancy and constructions of symmetrical uniform designs. J Complex 21:757–771
Guo Y, Simpson JR, Pignatiello JJ (2009) Optimal foldover plans for mixed-level fractional factorial designs. Qual Reliab Eng Int 25:449–466
Hickernell FJ (1998) A generalized discrepancy and quadrature error bound. Math Comput 67:299–322
Lei YJ, Ou ZJ, Qin H, Zou N (2012) A note on lower bound of centered \(L_2\)-discrepancy on combined designs. Acta Math Sin 28(4):793–800
Lei YJ, Qin H, Zou N (2010) Some lower bounds of centered \(L_2\)-discrepancy on foldover designs. Acta Math Sci 30A(6):1555–1561
Li F, Jacroux M (2007) Optimal foldover plans for blocked \(2^{m-k}\) fractional factorial designs. J Stat Plan Inference 137:2439–2452
Li H, Mee RW (2002) Better foldover fractions for resolution III \(2^{k-p}\) designs. Technometrics 44:278–283
Li PF, Liu MQ, Zhang RC (2005) Choice of optimal initial designs in sequential experiments. Metrika 61(2):127–135
Li W, Lin DKJ (2003) Optimal foldover plans for two-level fractional factorial designs. Technometrics 45:142–149
Li W, Lin DKJ, Ye KQ (2003) Optimal foldover plans for non-regular orthogonal designs. Technometrics 45:347–351
Liu MQ, Fang KT, Hickernell FJ (2006) Connections among different criteria for asymmetrical fractional factorial designs. Stat Sin 16(4):1285–1297
Ma CX, Fang KT (2001) A note on generalized aberration factorial designs. Metrika 53:85–93
Montgomery DC, Runger GC (1996) Foldover of \(2^{k-p}\) resolution IV experimental designs. J Qual Technol 28:446–450
Ou ZJ, Chatterjee K, Qin H (2011) Lower bounds of various discrepancies on combined designs. Metrika 74:109–119
Ou ZJ, Qin H, Cai X (2014) A lower bound for the wrap-around \(L_2\)-discrepancy on combined designs of mixed two- and three-level factorials. Commun Stat 43:2274–2285
Ou ZJ, Qin H, Cai X (2015) Optimal foldover plans of three level designs with minimum wrap-around \(L_2\)-discrepancy. Sci China Ser A 58:1537–1548
Qin H, Chatterjee K, Ou ZJ (2013) A lower bound for the centered \(L_2\)-discrepancy on combined designs under the asymmetric factorials. Statistics 47:992–1002
Sun FS, Chen J, Liu MQ (2011) Connections between uniformity and aberration in general multi-level factorials. Metrika 73(3):305–315
Wang B, Robert GM, John FB (2010) A note on the selection of optimal foldover plans for 16- and 32-run fractional factorial designs. J Stat Plan Inference 140:1497–1500
Wu CFJ, Hamada M (2009) Experiments: planning, analysis, and optimization, 2nd edn. Wiley, New York
Ye KQ, Li W (2003) Some properties of blocked and unblocked foldover of \(2^{k-p}\) designs. Stat Sin 13:403–408
Zhou YD, Fang KT, Ning JH (2012) Constructing uniform designs: a heuristic integer programming method. J Complex 28:224–237
Zhou YD, Fang KT, Ning JH (2013) Mixture discrepancy for quasi-random point sets. J Complex 29:283–301
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
Acknowledgements
The authors thank the Editor in Chief and the referees for their helpful comments. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11201177, 11271147, 11561025), China Postdoctoral Science Foundation (Grant No. 2013M531716), Scientific Research Plan Item of Hunan Provincial Department of Education (Grant No. 14B146).
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Ou, Z., Qin, H. Optimal foldover plans of asymmetric factorials with minimum wrap-around \(L_2\)-discrepancy. Stat Papers 60, 1699–1716 (2019). https://doi.org/10.1007/s00362-017-0892-x
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DOI: https://doi.org/10.1007/s00362-017-0892-x
Keywords
- Asymmetric fractional factorial
- Combined design
- Foldover plan
- Lower bound
- Wrap-around \(L_2\)-discrepancy