New Lower Bounds For Lee Discrepancy And Wrap-around L₂-discrepancy On Two And Three Mixed Levels Factorials

Lee-discrepancy(Zhou, Ning and Song,2008) and wrap-around L2-discrepancy (Hickernell,1998) have been utilized as measures for assessing and comparing unifor-mity of factorial designs. There have been many literatures to explore the potential application of these two discrepancies in fractional factorial designs. Recently,the issue of lower bounds for these discrepancies has been considerable interest. On the one hand, The accurate lower bound of Lee-discrepancy or wrap-around L2-discrepancy can be used to verify the uniformity of known fractional factorial designs. For the discrepancy value of a known fractional factorial design, the closer it is to the lower bound of discrepancy, then the better the uniformity of this design. On the other hand, the lower bound of discrepancy can be used as reference value of stochastic optimization algorithm to search and structure uniform designs. Study on the lower bound of discrepancy, therefore, is a meaningful subject.This paper focuses on lower bounds for Lee discrepancy and wrap-around L2discrepancy on two and three mixed levels factorials, We give a new lower bound of Lee discrepancy and wrap-around L2discrepancy, respectively. Compared with lower bounds in the literature, our lower bounds are more precise and more strict in some design class.Denote are integers such that P2and q2are integers such thatMain results of this paper are given as follows:Theorem4.1For any design D∈U(n;2S1×3S2), we have (LD(D))2≥LB2(LD(D)), if p1≥q2, we have if p1≤q2, we have whereTheorem5.1For any design D∈U(n;2S13S2), we have (WD2(d))2≥LB2(WD(D)), if p1> q2,we have if P1≤q2,we havewhere...