## The Annals of Statistics

### $D$-Optimum Weighing Designs

#### Abstract

For the problem of weighing $k$ objects in $n$ weighings $(n \geq k)$ on a chemical balance, and certain related problems, we obtain new results and list the designs which have been proved $D$-optimum up to this time. While some of these optimality results have been known for some time, others are fairly recent. In particular, in the most difficult case $n \equiv 3(\operatorname{mod} 4)$ we prove a result characterizing optimum designs when $n \geq 2k - 5$. In addition, by a combination of theoretical bounds and computer search we find previously unknown optimum designs in the cases $(k, n) = (9, 11), (11, 15)$, and (12, 15), and establish the optimality of Mitchell's (10, 11) design. In some cases the optimum $X'X$ is not unique. Thus, we find two optimum $X'X$'s for the (6, 7), (8, 11), (10, 11), and (10, 15) cases. As a consequence of these results and other constructions, $D$-optimum designs are now known in all cases $k \leq 12$ (for all $n \geq k$), and in many other cases. Essentially complete listings for all $n \geq k$ had been given previously only for $k \leq 5$.

#### Article information

Source
Ann. Statist., Volume 8, Number 6 (1980), 1293-1306.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345202

Digital Object Identifier
doi:10.1214/aos/1176345202

Mathematical Reviews number (MathSciNet)
MR594646

Zentralblatt MATH identifier
0466.62066

JSTOR
links.jstor.org

Subjects
Primary: 62K15: Factorial designs
Secondary: 05B20: Matrices (incidence, Hadamard, etc.)

#### Citation

Galil, Z.; Kiefer, J. $D$-Optimum Weighing Designs. Ann. Statist. 8 (1980), no. 6, 1293--1306. doi:10.1214/aos/1176345202. https://projecteuclid.org/euclid.aos/1176345202