Abstract
Uniform design is one of the most frequently used designs of experiment, and all factors are usually regarded as equally important in the existing literature of uniform design. If some prior information of certain factors is known, the potential importance of factors should be distinguished. In this paper, by assigning different weights to factors with different importance, the weighted wrap-around \(L_2\)-discrepancy is proposed to measure the uniformity of design when some prior information of certain factors are known. The properties of weighted wrap-around \(L_2\)-discrepancy are explored. Accordingly, the weighted generalized wordlength pattern is proposed to describe the aberration of these kinds of designs. The relationship between the weighted wrap-around \(L_2\)-discrepancy and weighted generalized wordlength pattern is built, and a lower bound of weighted wrap-around \(L_2\)-discrepancy is obtained. Numerical results show that both weighted wrap-around \(L_2\)-discrepancy and weighted generalized wordlength pattern are precisely to capture the difference of importance among the columns of design.
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Acknowledgements
The authors greatly appreciate helpful suggestions of Editor-in-Chief and the referees. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11961027, 12161040, 11701213), Hunan Provincial Natural Science Foundation of China (Grant Nos. 2021JJ30550, 2020JJ4497), Scientific Research Plan Item of Hunan Provincial Department of Education (Grant No. 19A403), Postgraduate Scientific Research Innovation Item of Jishou University (Grant No. JGY201931).
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Luo, B., Li, H., Wei, Y. et al. Uniform design with prior information of factors under weighted wrap-around \(L_2\)-discrepancy. Comput Stat 37, 2717–2739 (2022). https://doi.org/10.1007/s00180-022-01193-9
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DOI: https://doi.org/10.1007/s00180-022-01193-9