Optimality of some two-associate-class partially balanced incomplete- block designs.(English)Zbl 0741.62071

Summary: Let $${\mathcal D}_{v,b,k}$$ be the set of all the binary equireplicate incomplete-block designs for $$v$$ treatments in $$b$$ blocks of size $$k$$. It is shown that if $${\mathcal D}_{v,b,k}$$ contains a connected two-associate- class partially balanced design $$d^*$$ with $$\lambda_ 2=\lambda_ 1\pm 1$$ which has a singular concurrence matrix, then it is optimal over $${\mathcal D}_{v,b,k}$$ with respect to a large class of criteria including the $$A$$, $$D$$ and $$E$$ criteria. The dual of $$d^*$$ is also optimal over $${\mathcal D}_{b,v,r}$$ with respect to the same criteria, where $$r=bk/v$$.
The result can be applied to many designs which were not previously known to be optimal. In another application, the second author’s [J. Stat. Plann. Inference 18, 299-312 (1988; Zbl 0639.62066)] conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.

MSC:

 62K05 Optimal statistical designs 62K10 Statistical block designs 05B05 Combinatorial aspects of block designs

Zbl 0639.62066
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