Lower bound of centered L2L2-discrepancy for mixed two and three levels UU-type designs

• This paper gives a new analytical expression of the centered L2L2-discrepancy for mixed two and three levels UU-type designs.
• The first lower bound to the centered L2L2-discrepancy for UU-type designs with mixed two and three-level is obtained.
• Our result covers the partial work of Fang et al. (2005) and Chatterjee et al. (2012) on two-level or three-level UU-type designs as two special cases.

In this paper, we investigate a new analytical expression of the centered L2L2-discrepancy measure of uniformity for mixed two and three-level UU-type designs in depth. Based on this new formulation, we present a new lower bound to the centered L2L2-discrepancy for UU-type designs with mixed two and three-level, which can be used as a benchmark in searching uniform UU-type designs. We also describe a necessary condition for the existence of uniform designs meeting this lower bound. For illustration of the usage of our theoretical results, a catalog of lower bounds for UU-type designs in U(n;2m1×3m2)U(n;2m1×3m2) is tabulated, where 0≤m1≤100≤m1≤10, 0≤m2<250≤m2<25 and 6≤n<606≤n<60.