Uniformity and projection uniformity of combined designs

Abstract

The purpose of this paper is to study the issue of employing the uniformity criterion measured by the mixture discrepancy to assess the optimal foldover plans for q-level factorials. The average mixture discrepancy and the average projection mixture discrepancy based on the level permutation method are respectively defined for combined designs, and the optimal foldover plan in terms of the overall uniformity and the uniformity of each dimension are also explored. The tight lower bounds of the average mixture discrepancy and the uniformity pattern under the general foldover plan are obtained respectively, which can be used as a benchmark for searching optimal foldover plans. Some illustrative examples are provided to show the theoretical results.

Keywords:

• Average mixture discrepancy
• average projection mixture discrepancy
• combined designs
• lower bound
• optimal foldover plans
• uniformity pattern

MSC:

• 62K15
• 62K05
• 62K99

Acknowledgments

The authors sincerely thank the editor, the associate editor, and the two referees for their comments, which have led to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11961027, 12161040), the Natural Science Foundation of Hunan Provincial (Grant No. 2021JJ30550), the Discipline Coordination Construction Project of Zhongnan University of Economics and Law (Grant No. XKHJ202125) and the Fundamental Research Funds for the Central Universities of Zhongnan University of Economics and Law (202211307).