Lower bounds of the wrap-around L2L2-discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L2L2-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L2L2-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L2L2-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.