Space-filling Property Of Extended Designs And Fractional Factorials

With the development of science and technology, many new unsolved problems have been arising, which bring new opportunities and challenges to experimental designs, see Bates et al. (1996). How to choose appropriate designs and optimal designs in some sense among the alternative designs has been focused on by the theoretic researches and practical applications. Therefore, statisticians have proposed many criteria to compare and select optimal designs. Among these criteria, the uniformity criterion and minimum aberration criterion have excellent properties and are adopted widely, see Fang, Li and Sudjianto (2006) and Xu (2003).Facing a complex research object, a natural design strategy is to follow with the sequential experiment. Experimenters can conduct experiment according to a selected design in advance, while during or after this experiment they need to do some additional complementary experiment on the basis of an another design. So the selected design of the whole stage of this experiment was composed of the former and the latter design, we call it as the extended design. This paper studies the property of space-filling of extended designs, which intends to provide some necessary theoretic support of their applications. The study of this paper announces that when uniform designs are chosen as original designs, and nearly uniform designs also are chosen as additional designs, that strategy not only reduces the damage of the uniformity of the later to the former extensively, but also replies a good uniformity to the extended designs. With regards to the wrap-around L2 discrepancy and Lee discrepancy, we obtains lower bounds of the discrepancy of extended designs, respectively. These lower bounds can be served as a benchmark to compare the uniformity among different designs, and also improve the efficiency of computer search uniform extended designs. Inspired by uniform extended designs, we prove that the Lee discrepancy value of designs can be expressed by a quadratic form of their frequency vector, and obtain a link between a general mixed designs and its complementary design under Lee discrepancy.Cheng and Wu (2001) found that the level permutation of one or a lot of factors can change the geometrical structure of designs when the number of level is bigger than two, which lead to the change of statistical properties of these designs. Recently, the space-filling property of high level fractional factorial designs has been more and more focused by researchers, refer to Tang, Xu and Lin (2012), Zhou and Xu (2014). This paper also studies the space-filling property of minimum moment aberration designs. Our conclusion suggests that level permutation can not only improve the space-filling property of designs, but also remain their minimum moment aberration. We also give a link between the average discrepancy basis on kernel function and orthogonality, which provids a further statistical justification for the average discrepancy.