A lower bound for the centred L₂-discrepancy on combined designs under the asymmetric factorials

Abstract

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.

Keywords:

• asymmetrical fractional factorial
• centred L ₂-discrepancy
• combined design
• foldover plan
• lower bound

MSC: :

• 62K15
• 62K10
• 62K99

Acknowledgements

The authors would like to thank the Associate Editor and two Referees for their valuable comments and suggestions. This work was partially supported by SRFDP (No. 20090144110002), the Special Fund for Basic Scientific Research of Central Colleges and Scientific Research Plan Item of Hunan Provincial Department of Education (No. 10C1091).