## The Annals of Statistics

- Ann. Statist.
- Volume 19, Number 3 (1991), 1667-1671.

### Optimality of Some Two-Associate-Class Partially Balanced Incomplete-Block Designs

#### Abstract

Let $\mathscr{D}_{\upsilon,b,k}$ be the set of all the binary equireplicate incomplete-block designs for $\upsilon$ treatments in $b$ blocks of size $k$. It is shown that if $\mathscr{D}_{\upsilon,b,k}$ contains a connected two-associate-class partially balanced design $d^\ast$ with $\lambda_2 = \lambda_1 \pm 1$ which has a singular concurrence matrix, then it is optimal over $\mathscr{D}_{\upsilon,b,k}$ with respect to a large class of criteria including the $A,D$ and $E$ criteria. The dual of $d^\ast$ is also optimal over $\mathscr{D}_{b,\upsilon,r}$ with respect to the same criteria, where $r = bk/\upsilon$. The result can be applied to many designs which were not previously known to be optimal. In another application, Bailey's (1988) conjecture on the optimality of Trojan squares over semi-Latin squares is confirmed.

#### Article information

**Source**

Ann. Statist., Volume 19, Number 3 (1991), 1667-1671.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176348270

**Digital Object Identifier**

doi:10.1214/aos/1176348270

**Mathematical Reviews number (MathSciNet)**

MR1126346

**Zentralblatt MATH identifier**

0741.62071

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62K05: Optimal designs

Secondary: 05B05: Block designs [See also 51E05, 62K10]

**Keywords**

Optimal design partially balanced incomplete-block design regular graph design strongly regular graph design

#### Citation

Cheng, C.-S.; Bailey, R. A. Optimality of Some Two-Associate-Class Partially Balanced Incomplete-Block Designs. Ann. Statist. 19 (1991), no. 3, 1667--1671. doi:10.1214/aos/1176348270. https://projecteuclid.org/euclid.aos/1176348270