Researches In The Connection Among Several Important Criteria In Factorial Desgins

There are many different kinds of criteria to assess and compare the goodness of factorial designs. Generalized minimum aberration(GMA) criterion and minimum moment aberration(MMA) criterion are the two important criteria among them. Xu(2003) showed that the power moments were linear combinations of generalized word-length patterns. A new analytic relation between GMA and MMA is built by using the linear combinations of power moments to describe generalized wordlength patterns. Thereby , The equivalence relation between the two criteria is further illustrated .In recent years, uniformity has been utilized as a measure to compare and assess factorial designs. Uniformity is a geometry concept that is relative to uniform design(Fang and Wang ,1994; Fang et al ,2000). Uniform designs spread their experimental points evenly throughout the design space. Recently, a fruitfull criterion, called discrete discrepancy, was proposed by Fang, Lin and Liu(2003),Qin and Fang (2004) and Qin and Li(2006), and was applied to be a new measure of uniformity. We discussed the connections between MMA and discrete discrepancy /the centered L₂—discrepancy/the wrap-around L₂—discrepancy, then show a further link between uniformity and MMA.The main results of this paper are given as follows:Theorem 1 For d∈(N, sⁿ) and j = 1,…, n whereH_i= {1,…, n}/{n + 1 - i}.Theorem 2 Supposed d∈(N, sⁿ), Thenwhere a and b are constants and a > b > 0; X_ij was defined in Theorem 1.Theorem 3 Supposed d∈(N, sⁿ), we have(i) when s = 2, a = 5/4, b=1, (ii)when s = 2, a = 3/2, b = 5/4,(iii) when s = 3, a =3/2,6 = 23/18,wherewas defined in Theorem 1.