Uniformity In Terms Of The Symmetric L₂-discrepancy Of Double Designs

Two-level fractional factorial designs have been widely used in industry, agriculture and scientific experiments. For constructing two-level fractional factorial designs, a new method called doubling method has been recently used. In particular, in constructing those of resolution IV, doubling is a simple but very powerful method (Chen and Cheng, 2006; Xu and Cheng, 2006). Suppose X is a two-level fractional factorial design, in which two levels are denoted by+1 and - 1. The double of X is the following matrixIn this paper, we study double designs in view of uniformity. We use the symmetric L₂-discrepancy as the measure of uniformity (Ma et al, 2002; Wang et al, 2006), and provide some lower bounds of double designs in the sense of the symmetric L₂-discrepancy. Using these lower bounds, we can value uniformity of double designs. Furthermore, a close connection between uniformity of double design D(X) and that of the initial design X is also given. In addition, we also obtain results connecting the symmetric L₂-discrepancy of D(X) and the generalized wordlength pattern (Xu and Wu, 2002) of X. We conclude that if X has less aberration, then D(X) has lower symmetric L₂-discrepancy, i.e., D(X) has better uniformity.Main results of this paper are given as follows:Theorem 1 Let X∈U(n; 2^k), we have [SD₂(D(X))]²â‰¥L_SD2^c(2; n, k),where L_SD2^c(2; n,k) = C + 5^k/2^k+1 + 1/(2n²) and S_n,m,2 is the residual of n(mod 2^m), C = (4/3)^2k - 2(11/8)^2k + 2^k-1Theorem 2 Let X∈U(n; 2^k), we have [SD₂(D(X))]²â‰¥L_SD2^r(2; n, k), where L_SD2^r(2; n, k) = C + (n - 1)4^θ(1 + 3f)/(2n) + 4^k/(2n), λ= k(n - 2)/[2(n - 1)],λ=θ+ f,θis the largest integer contained in A, f =λ-θ,C = (4/3)^2k - 2(11/8)^2k + 2^k-1.Theorem 3 Let X∈V(n; 2^k), we havewhere C = (4/3)^2k - 2(11/8)^2k + 2^k-1.