Lee discrepancy and its applications in experimental designs

Various discrepancies have been defined in uniform designs, such as centered L2-discrepancy, wrap-around L2-discrepancy and discrete discrepancy. Among them the discrete discrepancy can explore relationships among uniform designs, fractional factorial designs, and combinational designs. However, the discrete discrepancy is mainly good for two-level factorial designs. In this paper, a new discrepancy based on the Lee distance, Lee discrepancy, is proposed and its computational formula is given. The Lee discrepancy can expand the relationships between the discrete discrepancy and some criteria for factorial designs with multiple levels. Some lower bounds of the Lee discrepancy for symmetrical and asymmetrical designs are given, and some connections between the Lee discrepancy and the generalized minimum aberration are considered. Finally, relationships between the Lee discrepancy and majorization framework are also considered.