The Annals of Mathematical Statistics

Some Optimum Weighing Designs

Damaraju Raghavarao

Full-text: Open access


Suppose we are given $N$ objects to be weighed in $N$ weighings with a chemical balance having no bias. Let $x_{ij} = 1$ if the $j$th object is placed in the left pan in the $i$th weighing, $= -1$ if the $j$th object is placed in the right pan in the $i$th weighing, $= 0$ if the $j$th object is not weighed in the $i$th weighing. The $N$th order matrix $X = (x_{ij})$ is known as the design matrix. Also let $y_i$ be the result recorded in the $i$th weighing, $\epsilon_i$ be the error in this result and $w_j$ be the true weight of the $j$th object, so that we have the $N$ equations $$x_{i1}w_1 + x_{i2}w_2 + \cdots + x_{iN}w_N = y_i + \epsilon_i, i = 1, \cdots, N.$$ We assume $X$ to be a non-singular matrix. The method of Least Squares or theory of Linear Estimation gives the estimated weights $(\hat w_i)$ by the equation $$\hat w = (X'X)^{-1}X'Y,$$ where $Y$ is the column vector of the observations and $\hat w$ is the column vector of the estimated weights. If $\sigma^2$ is the variance of each weighing, then $$\operatorname{Var} (\hat w) = (X'X)^{-1}\sigma^2 = (c_{ij})\sigma^2,$$ where $(c_{ij})$ is the inverse matrix of $(X'X)$. An expository article reviewing the work done in weighing designs is given by Banerjee [2]. Kishen [4] treats the reciprocal of the increase in variance resulting from the adoption of any design other than the most efficient design, with mean variance $\sigma^2/N$, as the efficiency of the design. This efficiency can be measured by $$1/\sum^N_{i=1} c_{ii}.$$ Mood [5] gives an alternative definition for the best weighing design. In his view the best weighing design should give the smallest confidence region in the $\hat w_i(i = 1, \cdots, N)$ space for the estimates of the weights. Hence a design will be called best if the determinant of the matrix $(c_{ij})$ is minimised. In this paper we follow Kishen's definition in obtaining the best weighing designs. Hotelling [3] proved that for the best weighing design $c_{ii} = 1/N$ and $c_{ij} = 0 (i \neq j)$. The weighing designs for which $c_{ii} = 1/N$ and $c_{ij} = 0$ are best in the sense of both Kishen and Mood. Later Mood proved that the above property is satisfied by Hadamard matrices. Plackett and Burman [6] have constructed Hadamard matrices, $H_N$, up to and including $N = 100$, excepting $N = 92$. It may be remarked here that a necessary condition for the existence of $H_N$ is $N \equiv 0 (\mod 4)$, with the exception of $N = 2$. It is not known whether this condition is sufficient or not. In this paper, the best weighing designs are obtained in the cases (i) $N$ is odd and (ii) $N \equiv 2 (\mod 4)$ subject to the conditions: i) The variances of the estimated weights are equal; and ii) The estimated weights are equally correlated. The 2nd condition here is the same as that of Banerjee [1].

Article information

Ann. Math. Statist., Volume 30, Number 2 (1959), 295-303.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Raghavarao, Damaraju. Some Optimum Weighing Designs. Ann. Math. Statist. 30 (1959), no. 2, 295--303. doi:10.1214/aoms/1177706253.

Export citation