标题： 最优BIBD扩展设计 显示英文标题

标题： 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，它们在广泛的最优性标准下是最佳的，但当BIBDs不存在时，不清楚应选择哪些设计来适应v、b和k的组合。 1992年，Cheng表明，对于足够大的b，根据常用标准（包括A-和D-标准）最优的设计必须在(M.S)-最优设计中找到。 特别是，这一结果证实了John和Mitchell在1977年关于正则图设计（RGDs）最优性的猜想，在区组数量较大的情况下成立。 我们研究通过反复添加BIBD的区组来扩展已知的最优二元设计的影响，并确定区组数量的边界，使得这些BIBD扩展的设计是最优的。 特别是，我们将研究k=2且b=v-1和b=v的情况；在这些情况下，A-和D-最优设计并不相同，但我们证明在添加BIBD的区组后，这种情况会发生变化，同一设计在扩展设计集合中成为A-和D-最优的。 最后，我们描述那些产生A-和D-最优扩展设计的RGDs，并将关于群可分设计的D-最优性的结果扩展到BIBD扩展设计中的A-和D-最优性。 显示英文摘要

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.