{"title":"跨两方块设计的最优性和构造","authors":"Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto","doi":"10.1007/s00184-024-00963-3","DOIUrl":null,"url":null,"abstract":"<p>We consider a statistical problem to estimate variables (effects) that are associated with the edges of a complete bipartite graph <span>\\(K_{v_1, v_2}=(V_1, V_2;E)\\)</span>. Each data is obtained as a sum of selected effects, a subset of <i>E</i>. To estimate efficiently, we propose a design called Spanning Bipartite Block Design (SBBD). For SBBDs such that the effects are estimable, we proved that the estimators have the same variance (variance balanced). If each block (a subgraph of <span>\\(K_{v_1, v_2}\\)</span>) of SBBD is a semi-regular or a regular bipartite graph, we show that the design is A-optimum. We also show a construction of SBBD using an (<span>\\(r,\\lambda \\)</span>)-design and an ordered design. A BIBD with prime power blocks gives an A-optimum semi-regular or regular SBBD.</p>","PeriodicalId":49821,"journal":{"name":"Metrika","volume":"84 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimality and constructions of spanning bipartite block designs\",\"authors\":\"Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto\",\"doi\":\"10.1007/s00184-024-00963-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider a statistical problem to estimate variables (effects) that are associated with the edges of a complete bipartite graph <span>\\\\(K_{v_1, v_2}=(V_1, V_2;E)\\\\)</span>. Each data is obtained as a sum of selected effects, a subset of <i>E</i>. To estimate efficiently, we propose a design called Spanning Bipartite Block Design (SBBD). For SBBDs such that the effects are estimable, we proved that the estimators have the same variance (variance balanced). If each block (a subgraph of <span>\\\\(K_{v_1, v_2}\\\\)</span>) of SBBD is a semi-regular or a regular bipartite graph, we show that the design is A-optimum. We also show a construction of SBBD using an (<span>\\\\(r,\\\\lambda \\\\)</span>)-design and an ordered design. A BIBD with prime power blocks gives an A-optimum semi-regular or regular SBBD.</p>\",\"PeriodicalId\":49821,\"journal\":{\"name\":\"Metrika\",\"volume\":\"84 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-04-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Metrika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00184-024-00963-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Metrika","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00184-024-00963-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Optimality and constructions of spanning bipartite block designs
We consider a statistical problem to estimate variables (effects) that are associated with the edges of a complete bipartite graph \(K_{v_1, v_2}=(V_1, V_2;E)\). Each data is obtained as a sum of selected effects, a subset of E. To estimate efficiently, we propose a design called Spanning Bipartite Block Design (SBBD). For SBBDs such that the effects are estimable, we proved that the estimators have the same variance (variance balanced). If each block (a subgraph of \(K_{v_1, v_2}\)) of SBBD is a semi-regular or a regular bipartite graph, we show that the design is A-optimum. We also show a construction of SBBD using an (\(r,\lambda \))-design and an ordered design. A BIBD with prime power blocks gives an A-optimum semi-regular or regular SBBD.
期刊介绍:
Metrika is an international journal for theoretical and applied statistics. Metrika publishes original research papers in the field of mathematical statistics and statistical methods. Great importance is attached to new developments in theoretical statistics, statistical modeling and to actual innovative applicability of the proposed statistical methods and results. Topics of interest include, without being limited to, multivariate analysis, high dimensional statistics and nonparametric statistics; categorical data analysis and latent variable models; reliability, lifetime data analysis and statistics in engineering sciences.