【作者】 王静；

【导师】 岳荣先；

【作者基本信息】 上海师范大学 ， 概率论与数理统计， 2021， 博士

【摘要】 正交表在统计上是不可或缺的,它主要用于试验设计,这就意味着正交表在各个领域都十分重要.自Rao(1947)介绍了正交表的结构和应用之后,许多组合数学家和统计学家都致力于正交表的研究,并提出了许多巧妙的构造方法.在试验设计中,由于受费用的限制,我们可能不使用行数较大的正交表.随着研究的不断深入,高强度正交表也可用于其他领域,如量子信息,计算机科学和密码学等领域.由于要考虑多列正交性,因此构造高强度的正交表更具有挑战性.到目前为止,已知的高强度正交表仍是稀缺的,这也大大限制了它们的应用.Hedayat等(1999)提出一个研究问题:有待开发更好的方法和工具以构造强度t ≥ 3的非对称正交表.因此,构造高强度正交表是迫切需要的,特别是含有非素数幂水平的正交表.构造具有最少行数的正交表(紧的正交表)以及最大因子数的正交表一直是人们关注的焦点.本文研究了利用正交分划和分层加法两类方法构造高强度非对称正交表.这些方法不仅简单,而且由于它们不依赖于有限域和差阵,所以对构造任意强度、水平数和各种行数的正交表非常有用.运用这些构造方法可获得大量的新的高强度正交表和其他正交表类,其中包括含有非素数幂水平的正交表,紧的正交表和最大可能因子数的正交表.附录列举出一部分构造的新的正交表以便读者使用.众所周知,确定正交表的存在性是构造该正交表的重要前提.对于给定的参数,研究它们的存在性具有实际和理论意义.然而关于强度大于2的非对称正交表的存在性的相关研究相对较少,特别是紧的正交表.在一些优良性准则下,正交表作为一种部分因子设计是最优的.当可用的试验次数略大于正交表的行数时,通过将额外的行连接到正交表上来构造部分因子设计,因此本文研究了正交表加p行设计的D-最优性.结合Rao不等式,我们分别定义了偶数强度和奇数强度的正交表对应的扩展矩阵.这些扩展矩阵是解决问题的重要突破口.运用扩展矩阵的性质,本文不仅讨论了高强度紧的正交表的存在的必要条件并列举一些紧的正交表不存在的例子,而且研究了强度t的正交表加p行设计达到D-最优时需要满足的条件.特别地,当t=2时,本文对Bird和Street(2016)提出的问题给出了积极的回答。更多还原

【Abstract】 Orthogonal arrays(OAs)are indispensable in statistical and are mainly used for design of experiments,which means that OAs are important in all fields.Since Rao(1947)introduced the structure and application of OAs,many combinatorial mathematicians and statisticians have devoted themselves to the research of OAs,and obtained many ingenious construction methods.In the experimental design,due to the limitation of cost,we may not use the OAs with large runs.However,along with the deepening of research,OAs of high strength can also be applied in other fields,such as quantum information,computer science and cryptography.It is more challenging to construct OAs of high strength because the orthogonality of multiple columns.So far,existing asymmetric OAs of high strength are scarce,also limiting their applications.Hedayat et al.(1999)proposed Research Problem:develop better methods and tools for the construction of mixed OAs with strength t≥3.Therefore,there is a need for the construction of high strength OAs，especially some OAs having factors whose numbers of levels are nonprime powers.Constructing OAs with the least number of runs(tight OAs)and OAs with the maximum numbers of factors is always of high interest.In this thesis,we study two class methods with orthogonal partition and juxtaposition to construct asymmetric OAs of high strength.Not only are the methods straightforward,but also they are useful for constructing OAs of arbitrary strengths,numbers of levels,and various sizes since they do not rely on the difference schemes and finite fields.A large number of new OAs and other families of OAs of high strength are constructed by using these construction methods,including OAs having factors whose numbers of levels are nonprime powers,tight OAs and OAs with the largest possible numbers of factors.Some new constructed OAs in Appendix are tabulated for reader’s use.It is well-known that determining the existence of OAs is an important prerequisite for constructing OAs.Their existence,for given values of the parameters,is of both practical and theoretical interest.However,relatively less work on the existence of asymmetric OAs of strength greater than two is available,especially tight OAs.OAs are optimal as a class of fractional factorial designs according to a range of optimality criteria.This makes it tempting to construct fractional factorial designs by adjoining additional runs to an OA when the number of runs available for the experiment is only slightly larger than the number in the OA,hence the D-optimality of OA plus p run designs is studied.In this thesis,combining with the Rao’s inequalities,some extended matrices corresponding to OAs of even strength and odd strength are respectively defined.These extended matrices are an important breakthrough to solve the problem.According to the properties of the extended matrix,we not only discuss the necessary conditions for the existence of tight OAs of high strength and illustrate some examples for the nonexistence of tight OAs,but also study the conditions that need to be satisfied when an OA of strength t plus p run design is D-optimal.Specially,when t=2,it gives a positive answer to the question proposed by Bird and Street(2016).更多还原