Lower bounds of projection weighted symmetric discrepancy on uniform designs

Abstract

A critical aspect of experimental designs is to determine the effective and efficient lower bounds of the discrepancy criterion in uniform designs. These lower bounds serve as benchmarks for measuring the design uniformity and for constructing uniform designs. Nowadays, symmetric discrepancy and projection weighted symmetric discrepancy are two commonly used discrepancy criteria. In this paper, we investigate the general lower bounds of these two discrepancies for symmetric multi-level designs and present sharp lower bounds for three-level designs, thereby complementing the existing lower bound theory of discrepancies in uniform designs. Several design examples are used to validate the theoretical results presented. Furthermore, we conduct two popular practical computer experiments to evaluate the performance of uniform designs based on these two discrepancies.