Uniformity in factorial designs with mixed levels

The uniformity can be utilized as a measure for comparing factorial designs. Fang and Mukerjee (Biometrika 87 (2000) 193–198) and Fang et al. (in: K.T. Fang, F.J. Hickernell, H. Niederreiter (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002) found links among uniformity in terms of some non-uniformity measures, orthogonality and aberration for regular symmetric factorials. In this paper we extend their results to asymmetric factorials by considering a so-called wrap-around L₂-discrepancy to evaluate the uniformity of factorials. Furthermore, a lower bound of wrap-around L₂-discrepancy is obtained for asymmetric factorials and two new ways of construction of factorial designs with mixed levels are proposed.