Constructing uniform designs under mixture discrepancy

The discrepancies have played an important role in quasi-Monte Carlo methods and uniform design. Zhou et al. (2013) proposed a new type of discrepancy, mixture discrepancy (MD), and showed that MD may be a better uniformity measure than wrap-around L2-discrepancy and centered L2-discrepancy. In this paper, some constructing methods for uniform designs under MD are shown. The relationship between MD and the generalized wordlength pattern for multi-level design is given, then the level permutation technique is shown as a useful tool to search uniform designs. Moreover, it is shown that MD can be represented by quadratic form and the global optimum solution of experimental design under MD is also given. Furthermore, by such quadratic form, the relationship between a design and its complementary design is shown, which can be used to construct uniform design with large size.