Existence of nested designs with block size five |
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| Institution: | 1. National Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Beijing 100083, China;2. Institute of Earth Sciences, China University of Geosciences, Beijing 100083, China;3. School of Earth Science and Resources, China University of Geosciences (Beijing), Beijing 100083, China;4. College of Resource, Hebei GEO University, 136 Huaiandong Road, Shijiazhuang 050031, Hebei Province, China;5. Department of Earth Science, University of Adelaide, Adelaide, SA 5005, Australia;1. Key Laboratory of Mineral Resources, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China;2. College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.;3. Geology Department, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario P7B 5E1, Canada;4. Institutions of Earth Science, Chinese Academy of Sciences, Beijing 100029, China;1. School of Earth Sciences, Lanzhou University, Lanzhou 730000, China;2. Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China;3. Department of Earth Sciences, Durham University, Durham DH1 3LE, UK;4. School of Earth Science and Mineral Resources, China University of Geosciences, Beijing 100083, China;5. Radiogenic Isotope Facility, School of Earth Sciences, The University of Queensland, Brisbane QLD 4072, Australia |
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| Abstract: | The block designs considered here are nested in the unified sense of Preece (Biometrika 54 (1967) 479–486) and Federer (in: T.A. Baneroft (Ed.), Statistical Papers in Honor of George W. Snedecor, 1972, pp. 91–114), that is, each block of the larger balanced incomplete block design contains several distinguished families of mutually disjoint sub-blocks, the sub-blocks of the same family belonging to the same system, such that each system forms the collection of blocks of some balanced incomplete block design. In this paper, it is shown that the necessary conditions for the existence of such designs are also sufficient for block size 5 and sub-block sizes 2 and 3. This, together with known results, implies the entire existence of such designs with block size 5 in general. |
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