| Statistics is a common used science, which explores the thinking and methodol-ogy of inferring the world through the data obtained by observing or collecting from the real world. While, experimental design, as a branch of statistics, studies how to efficiently observe and analyze an examined object by designing suitable experiments, which plays an important role for the development of statistics.When studying an object in the real world, there are often many dependent vari-ables which may affect the object as response variable. So a main subject in designing experiments is to consider the designs which involve many factors. For the operabil-ity of experiments, each of the factors is supposed to have several level settings and a factor level combination is also called a treatment. Running a set of treatments one by one is called a factorial experiment and the design consisting of the set of factor level combinations is called a factorial design. We call the design containing all the factor level combinations a full factorial design, otherwise call it a fractional factorial design. If the number of factors in a design is large, the run number of a full design is too large so that it is often impossible to carry out in practice because of the huge cost or time consumption. So fractional factorial experiments are more suitable in practice. For this reason, the most of investigations in the past of decades are focused onto consider fractional factorial designs and to select optimal one among them.Since the sixty of last century, the optimality theory of factorial design was de-veloped rapidly. Quite a few optimality criteria, such as maximum resolution (MR)[5], minimum aberration (MA)[29], clear effects (CE)[54] and maximum estimation capacity (MEC)[46], were successively proposed and widely applied in practice (see references [5,10,12,24,29,46,48,54,56,61]) Particularly, in2006, a new criterion, general minimum lower-order confounding (GMC), was proposed (see Zhang et al.[66]), which filled the gap that there lacked optimal designs for the experiments in which factors are not equally important. Different from the criteria before, they first introduced a new classi-fication pattern for regular designs, called aliased effect number pattern (AENP). The new pattern contains the minimum sufficient information about the confounding be-tween the effects in a design. Based on the AENP and effect hierarchy principle (EPH) they proposed the GMC criterion. The optimal designs selected by GMC criterion are called GMC designs. GMC designs have also been widely used in practice. It was proved that when there is some prior about the importance order of factors in an exper-iment (which appears very frequently in practice) the GMC design is the best choice for experimenter. So GMC designs are very important in practice. For convenience, we call the theory relating to AENP and GMC criterion a GMC theory.The GMC theory was developed quickly in recent years and many results of it have been obtained. For example, it was found that MA, CE and GMC criteria can be expressed as functions of AENP, which deeply reveal the essence, difference and relation of the existing criteria; also it was proved that in the case that only main effects and two factors interaction effects (2fi’s) are considered the MA and MEC criteria are equivalent (see [66] and[32]). Thus, GMC theory can be used to unify the existing crite-ria. AENP and GMC criterion of two-level case have been extended to s-level (where s is prime or power of prime) case (see paper[67]). On the construction of two-level regular GMC designs some significant results were obtained (see [34,64,21]). Several optimal blocked GMC criteria were established and some results on the construction of the several kinds of blocked GMC designs were also obtained (see [68,51,65,50,70]). Furthermore, GMC theory has been extended to split-plot design and robust parame-ter design (see [53] and [421); and it was also extended to nonregular factorial designs (see [22] and[74]), and so on. As an expansion of GMC theory, paper[73] introduced the concept of F-AENP for ranking columns of a regular design, and used it in GMC2n-m designs with5N/16<n≤N-1.Though lots of works on GMC theory have been finished as mentioned above, there are many important topics need to consider. For example, the construction of GMC2n-m designs for the parameter range n≤N/4, the construction of blocked B-B1-and B2-GMC designs for the parameter range n≤5N/16, and the construction of general s levels optimal GMC designs, and the arrange of factors to these designs. This thesis devotes to getting some answers to the above topics.In Chapter1, we first briefly introduce some development history and classifica-tion of experimental design. Then recall some optimality criteria, such as MR, MA, CE and MEC criteria. In Chapter2, we review the GMC theory and its development in recent years, containing some backgrounds, the concept of GMC criterion, the re-lationships with existing optimal criteria and the recent progressions development of GMC theory as well.In the later chapters we focus onto address the works we have done. Firstly, in Chapter3we introduce some works on the construction of GMC2n-m designs. For given q=n-m and experimental run N=2q, the number n of factors of a2n-m design can be q, q+1,..., N-1. As indicated above, Li, Zhao&Zhang[34], Zhang&Cheng[64] and Cheng&Zhang[21] have accomplished the construction of GMC2n-m designs for N/4+1≤n≤N-1. However, except for the trivial case m=0the construction of GMC2n-m designs for the remaining parameters q<n≤N/4is left for people to complete. Obviously, finishing the construction of GMC designs for this part of parameters is significantly important in both the theory and application. In this chapter we just address the works we have finished on it. First we prove that a design has GMC only if its defining contrast subgroup contains all the letters. Then through modifying the definition of Doubling and RC Yates order introduced in Zhang&Cheng[64], we simplify and unify the construction of GMC2n-m for N/4+1≤n≤5N/16in Zhang&Cheng[64] and Cheng&Zhang[21]. For the case of n<N/4+1, we propose a general algorithm to construct GMC2n-m designs. Based on the new algorithm, we obtain all the GMC2n-m designs for the parameters m≤4with any n and n=(2m-1)u+r, r-0,1,2,2m-3,2m-2with any nonnegative intege u. In the proofs of above results we get an interesting result:all the GMC designs obtained above are also MA designs except for the GMC29-4design.In Chapter4, we consider the computation of AENP. As we known, the AENP of a regular design describes the confounding structure between all the effects of it elaborately and explicitly. Hence computing the AENP of a design is an important issue. When N is large, directly calculating AENP will take a great amount of time. Thus, to find an easy method for computation is needed. In this chapter, based on the class partitions of Hq introduced in Zhou, Balakrishnan&Zhang, we get the distribution of all B2(d,γ)’s of GMC design d with n≥N/4+1, further obtain a simple formula of calculating#1C2and#2C2in AENP of a GMC design. By the formula the time of computing#1C2on a computer is even negligible, since it simply depends two parameters n and m without any matrix operation.The theory of blocked GMC criterion and corresponding designs is an important part of GMC theory. Zhang&Mukerjee, Wei, Li&Zhang and Zhang, Li&Wei established B-, B1-and B2-GMC criteria for blocked fractional factorial design respectively. Then, Tan&Zhang obtained all the B-GMC2n-m:2r designs with5N/16+1≤n≤N-1and r=1,2; Zhao et al. obtained all the B’-GMC2n-m:2r designs with5N/16+1≤n≤N-1and any r. But for the other ranges of parameters n and r the construction of B-, B1-and B2-GMC2n-m:2r designs has not been obtained yet. To finish the construction there are many works to do. In Chapter5, we propose a new method to construct B1-GMC2n-m:2r designs, which is based on the distribution of B2(d, y)’s of a GMC2n-m design d. By the new method, we not only obtain the construction results in Zhao et al. again, but also obtain the B’-GMC2n-m:2r designs with N/4+1≤n≤5N/16. Its construction results are simple and easy to use. |
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