A new lower bound for wrap-around \(L_2\)-discrepancy on two and three mixed level factorials (Q2339538)

This paper deals with the problem of lower bounds for the wrap-around \(L_2\)-discrepancy (written as \(\mathrm{WD}_2\) for short) on two and three level factorials which has been widely used to evaluate the uniformity of fractional factorial designs. The paper is divided into four sections. Section 1 is on introduction presenting the contributions made to this area by numerous researchers. It is pointed out that the lower bound for \(\mathrm{WD}_2\) can be used gainfully not only in searching for uniform U-type designs but also in validating that some good designs are in fact uniform. After presenting notations and preliminaries in Section 2, the next Section 3 presents some important lemmas and main results which provide us a new lower bound for wrap-around \(L_2\)-discrepancy. The authors provide four illustrative examples in Section 4, and using numerical simulation and examples that the new lower bound for \(\mathrm{WD}_2\) on asymmetrical factorial designs with two and three mixed levels behaves better than those which are already available in literature. (The following are the main three results: Result 1: For any design \(d\in U(n;2^{S_1}3^{S_2})\), \([\mathrm{WD}_2(d)]^2\geq LB_1\) Result 2: For any design \(d\in U(n;2^{S_1}3^{S_2})\), \(S=S_1+S_2\), we have \([\mathrm{WD}_2(d)]^2\geq LB_2\) Result 3: For any design \(d\in U(n;2^{S_1}3^{S_2})\), \(S=S_1+S_2\), we have \([\mathrm{WD}_2(d)]^2\geq LB\) where \(LB = \max[LB_1,LB_2]\)).