最优BIBD-扩展设计
Optimal BIBD-extended designs
摘要：
平衡不完全区组设计（BIBDs）是一类具有v个处理和b个大小为k的区组的设计，这些设计在广泛的最优性准则范围内是最优的，但不清楚当BIBDs不存在时，对于v，b和k的组合应选择哪些设计。1992年，Cheng证明了对于足够大的b，对于常用准则（包括A-和D-准则）最优的设计必须在（M.S）-最优设计中找到。特别地，这个结果证实了John和Mitchell在1977年关于在大量块的情况下正则图设计（RGDs）的最优性的猜想。我们调查的影响，通过重复添加的BIBD块和块的数量，使这些BIBD扩展的设计是最优的，找到边界的扩展已知的最优二进制设计。特别是，我们将研究k=2和b=v-1和b=v的设计：在这些情况下，A最优设计和D最优设计是不一样的，但我们表明，在添加BIBD的区组后，这种情况发生了变化，并且在扩展设计的集合中，相同的设计成为A最优设计和D最优设计。最后，我们将这些RGD，产生A-和D-最优的扩展设计和扩展的结果的D-最优的一组可分设计A-和D-最优的BIBD-扩展设计。
英文摘要：
Balanced incomplete block designs (BIBDs) are a class of designs with v treatments and b blocks of size k that are optimal with regards to a wide range of optimality criteria, but it is not clear which designs to choose for combinations of v, b and k when BIBDs do not exist. In 1992, Cheng showed that for sufficiently large b, the designs which are optimal with respect to commonly used criteria (including the A- and D- criteria) must be found among (M.S)-optimal designs. In particular, this result confirmed the conjecture of John and Mitchell in 1977 on the optimality of regular graph designs (RGDs) in the case of large numbers of blocks. We investigate the effect of extending known optimal binary designs by repeatedly adding the blocks of a BIBD and find boundaries for the number of block so that these BIBD-extended designs are optimal. In particular, we will study the designs for k=2 and b=v-1 and b=v: in these cases the A- and D-optimal designs are not the same but we show that this changes after adding blocks of a BIBD and the same design becomes A- and D-optimal amongst the collection of extended designs. Finally, we characterise those RGDs that give rise to A- and D-optimal extended designs and extend a result on the D-optimality of the a group-divisible design to A- and D-optimality amongst BIBD-extended designs.
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