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題名 The Construction and E-optimality of Linear Trend-Free Block Designs 作者 高建國 貢獻者 丁兆平<br>蔡風順
高建國關鍵詞 BIB design
Linear trend-free design
Nearly trend-free design
E-optimal日期 2002 上傳時間 10-五月-2016 18:56:14 (UTC+8) 摘要 Suppose there is a systematic effect or trend that influences the observations in addition to the block and treatment effects. The problem of experimental designs in the presence of trends was first studied by Cox (1951,1952). Bradley and Yeh (1980) define the concept of trend-free block designs, i.e., the designs in which the analysis of treatment effects are essentially the same whether the trend effects are present or not. If the trend effect within each blocks are the same and linear, Yeh and Bradley (1983) derive a simple necessary condition for designs to be linear trend-free, ri(k+1)≡0 (mod 2), 1≦i≦v, (1) where ri is the replication of treatment i, for 1≦i≦v, and k is block size. In case where a trend-free version does not exist Yeh et al. (1985) suggest the use of “ nearly trend-free version”. Chai (1995) pays attention to situations where (1) does not hold. He also shows that often, under these circumstances, a nearly linear trend-free design could be constructed. Designs that are derived by extending or deleting m disjoint and binary blocks from BIBD (v,b,k,r,λ)`s are considered. If the resulting designs have linear trend-free versions, by Constantine (1981), they are E-optimal designs with the corresponding classes. When k is even, however, it is impossible to have linear trend-free versions since not all the ri`s are even in such type of designs and (1) is violated. In this paper, we shall convert the designs to be nearly linear trend-free versions of them by permuting the treatment symbols within blocks, and investigate that the resulting designs remain to be E-optimal. 參考文獻 Agrawal, H. (1966). Some generalizations of distinct representatives with applications to statistical designs. Ann. Math. Statist. 37, 526-527. Agrawal, H. and Prasad, J. (1981). On the structure of incomplete block designs. Calcutta Statistical Association Bulletin 30, 65-76. Bradley, R.A. and Yeh, C.M. (1980). Trend-free block designs: theory. Ann. Statist. 8, 883-893. Chai, F.S. and D.Majumdar (1993). On the Yeh-Bradley conjecture on linear trend-free block design. Ann. Statist. 21, 2087-2097. Chai, F.S. (1995). Construction and optimality of nearly linear trend-free designs. J. Statist. Plann. Inference 48, 113-129. Chai, F.S. (1998). A note on generalization of distinct representatives. Statistics & Probability Letters 39, 173-177. Cochran, W.G. and Cox, G.M. (1957). Experimental Designs. 2nd edition. Wiley, New York. Connor, W.S., Jr. (1952). On the structure of balanced incomplete block designs. Ann. Math. Statist. 23, 57-71. Constantine, G.M. (1981). Some E-optimal block designs. Ann. Statist. 9, 886-892. Cox, D.R. (1951). Some systematic experimental designs. Biometrika 39, 312-323. Cox, D.R (1952). Some recent work on systematic experimental designs. J. Roy. Statist. Soc. Ser. B. 14, 211-219. Graybill, F.A. (1971). Introduction to Matrices with Applications in Statistics. Wadsworth Publishing, Belmont, CA. Hall, M. Jr. (1986). Combinatorial Theory. 2nd edition. Wiley, New York. Jacroux, M., Majumdar, D. and Shah, K.R. (1995). Efficiency block designs in the presence of trends. Statist. Sinica 5, 605-615. Jacroux, M., Majumdar, D. and Shah, K.R. (1997). On the determination and construction of optimal block designs in the presence of linear trends. J. Amer. Satist. Ass. 92, 375-383 Jacroux, M. (1998). On the determination and construction of E-optimal block designs in the presence of linear trends. J. Statist. Plann. Inference 67, 331-342. König, D. (1950). Theorie der Endlichen and Unendlichen Graphen 170-178. Chelsea, New York. Lin, M. and Dean, A.M. (1991). Trend-free block designs for varietal and factorial experiments. Ann. Statist. 19, 1582-1596. Raghavarao, D. (1971). Construction and Combinatorial Problems of Design of Experiments. Wiley, New York. Stufken, J. (1988). On the existence of linear trend-free block designs. Comm. Statist. Theory Methods 17, 3857-3863. Yeh, C.M. and Bradley, R.A. (1983). Trend-free block design; existence and construction results. Comm. Statist. Theory Methods 12, 1-24. Yeh, C.M., Bradley, R.A. and Notz, W.I. (1985). Nearly trend-free block designs. J. Amer. Statist. Assoc. 80, 985-992. 描述 博士
國立政治大學
統計學系
81354003資料來源 http://thesis.lib.nccu.edu.tw/record/#A2010000074 資料類型 thesis dc.contributor.advisor 丁兆平<br>蔡風順 zh_TW dc.contributor.author (作者) 高建國 zh_TW dc.creator (作者) 高建國 zh_TW dc.date (日期) 2002 en_US dc.date.accessioned 10-五月-2016 18:56:14 (UTC+8) - dc.date.available 10-五月-2016 18:56:14 (UTC+8) - dc.date.issued (上傳時間) 10-五月-2016 18:56:14 (UTC+8) - dc.identifier (其他 識別碼) A2010000074 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/96300 - dc.description (描述) 博士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計學系 zh_TW dc.description (描述) 81354003 zh_TW dc.description.abstract (摘要) Suppose there is a systematic effect or trend that influences the observations in addition to the block and treatment effects. The problem of experimental designs in the presence of trends was first studied by Cox (1951,1952). Bradley and Yeh (1980) define the concept of trend-free block designs, i.e., the designs in which the analysis of treatment effects are essentially the same whether the trend effects are present or not. If the trend effect within each blocks are the same and linear, Yeh and Bradley (1983) derive a simple necessary condition for designs to be linear trend-free, ri(k+1)≡0 (mod 2), 1≦i≦v, (1) where ri is the replication of treatment i, for 1≦i≦v, and k is block size. In case where a trend-free version does not exist Yeh et al. (1985) suggest the use of “ nearly trend-free version”. Chai (1995) pays attention to situations where (1) does not hold. He also shows that often, under these circumstances, a nearly linear trend-free design could be constructed. Designs that are derived by extending or deleting m disjoint and binary blocks from BIBD (v,b,k,r,λ)`s are considered. If the resulting designs have linear trend-free versions, by Constantine (1981), they are E-optimal designs with the corresponding classes. When k is even, however, it is impossible to have linear trend-free versions since not all the ri`s are even in such type of designs and (1) is violated. In this paper, we shall convert the designs to be nearly linear trend-free versions of them by permuting the treatment symbols within blocks, and investigate that the resulting designs remain to be E-optimal. zh_TW dc.description.tableofcontents 致謝辭 Abstract Contents 1. Introdution-----1 2. Notation and Preliminary Result-----5 3. Eigenvalue-----13 4. Construction Method-----21 5. Main Result-----96 References-----104 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#A2010000074 en_US dc.subject (關鍵詞) BIB design en_US dc.subject (關鍵詞) Linear trend-free design en_US dc.subject (關鍵詞) Nearly trend-free design en_US dc.subject (關鍵詞) E-optimal en_US dc.title (題名) The Construction and E-optimality of Linear Trend-Free Block Designs en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) Agrawal, H. (1966). Some generalizations of distinct representatives with applications to statistical designs. Ann. Math. Statist. 37, 526-527. Agrawal, H. and Prasad, J. (1981). On the structure of incomplete block designs. Calcutta Statistical Association Bulletin 30, 65-76. Bradley, R.A. and Yeh, C.M. (1980). Trend-free block designs: theory. Ann. Statist. 8, 883-893. Chai, F.S. and D.Majumdar (1993). On the Yeh-Bradley conjecture on linear trend-free block design. Ann. Statist. 21, 2087-2097. Chai, F.S. (1995). Construction and optimality of nearly linear trend-free designs. J. Statist. Plann. Inference 48, 113-129. Chai, F.S. (1998). A note on generalization of distinct representatives. Statistics & Probability Letters 39, 173-177. Cochran, W.G. and Cox, G.M. (1957). Experimental Designs. 2nd edition. Wiley, New York. Connor, W.S., Jr. (1952). On the structure of balanced incomplete block designs. Ann. Math. Statist. 23, 57-71. Constantine, G.M. (1981). Some E-optimal block designs. Ann. Statist. 9, 886-892. Cox, D.R. (1951). Some systematic experimental designs. Biometrika 39, 312-323. Cox, D.R (1952). Some recent work on systematic experimental designs. J. Roy. Statist. Soc. Ser. B. 14, 211-219. Graybill, F.A. (1971). Introduction to Matrices with Applications in Statistics. Wadsworth Publishing, Belmont, CA. Hall, M. Jr. (1986). Combinatorial Theory. 2nd edition. Wiley, New York. Jacroux, M., Majumdar, D. and Shah, K.R. (1995). Efficiency block designs in the presence of trends. Statist. Sinica 5, 605-615. Jacroux, M., Majumdar, D. and Shah, K.R. (1997). On the determination and construction of optimal block designs in the presence of linear trends. J. Amer. Satist. Ass. 92, 375-383 Jacroux, M. (1998). On the determination and construction of E-optimal block designs in the presence of linear trends. J. Statist. Plann. Inference 67, 331-342. König, D. (1950). Theorie der Endlichen and Unendlichen Graphen 170-178. Chelsea, New York. Lin, M. and Dean, A.M. (1991). Trend-free block designs for varietal and factorial experiments. Ann. Statist. 19, 1582-1596. Raghavarao, D. (1971). Construction and Combinatorial Problems of Design of Experiments. Wiley, New York. Stufken, J. (1988). On the existence of linear trend-free block designs. Comm. Statist. Theory Methods 17, 3857-3863. Yeh, C.M. and Bradley, R.A. (1983). Trend-free block design; existence and construction results. Comm. Statist. Theory Methods 12, 1-24. Yeh, C.M., Bradley, R.A. and Notz, W.I. (1985). Nearly trend-free block designs. J. Amer. Statist. Assoc. 80, 985-992. zh_TW
