Some new lower bounds to various discrepancies on combined designs

ABSTRACT

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this article, some new lower bounds of various discrepancies of combined designs, such as the centered, symmetric, and wrap-around L₂-discrepancies, are obtained, which can be used as a better benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.

KEYWORDS:

• Combined design
• Discrepancy
• Lower bound
• Optimal foldover plan
• Uniformity

MATHEMATICS SUBJECT CLASSIFICATION:

• 62K15
• 62K05
• 62K99

Acknowledgments

The authors greatly appreciate helpful suggestions of the referees and the editor-in-chief.

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11201177, 11271147), China Postdoctoral Science Foundation (Grant No. 2013M531716), and Scientific Research Plan Item of Hunan Provincial Department of Education (Grant Nos. 14B146).