Majorization framework for balanced lattice designs

Abstract

This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for assessing designs, which includes a stringent criterion of majorization via pairwise coincidences and flexible surrogates via convex functions. Classical orthogonality, aberration and uniformity criteria are unified by choosing combinatorial and exponential kernels. A construction method is also sketched out.Comment: Published at http://dx.doi.org/10.1214/009053605000000679 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some lower bounds of centered $$L_2$$ L 2 -discrepancy of $$2^{s-k}$$ 2 s - k designs and their complementary designs

“…A two-level design whose centered L 2 -discrepancy value achieves the lower bound in Theorem 1 is a uniform design, see Zhang et al (2005). In this section we present some illustrative examples to show our theoretical results.…”

“…Following the definition of Schur-exponential criterion in Zhang et al (2005), it is clear that (F; ϕ) is Schur-exponential criterion. Based on Theorem 1 in Zhang et al (2005), we can obtain that for the Schur-exponential criterion (F; ϕ), their lower bound is…”

“…Based on Theorem 1 in Zhang et al (2005), we can obtain that for the Schur-exponential criterion (F; ϕ), their lower bound is…”

“…The lower bound LCD 1 (s, 2) and LCD 2 (s, 2) can be achieved if and only if there are d γ word(s) with γ letters and d γ +1 word(s) with γ + 1 letters among the three words of the indicator function of F. Fang and Mukerjee (2000), Fang et al (2002), Fang et al (2003) and Zhang et al (2005) gave the lower bounds for the centered L 2 -discrepancy on two-level factorials, respectively. For 2 s−k designs and their complementary designs, numerical results which given in the next section show that our lower bound is better than the above lower bounds.…”

“…We also compare the new lower bound obtained in this paper to the existing results. We consider four existing lower bounds which are obtained in Fang and Mukerjee (2000), Fang et al (2002), Fang et al (2003) and Zhang et al (2005), respectively. For convenience, we denote these lower bounds as LC D F M , LC D FMM , LCD FLW and LC D ZFLS .…”