Some Optimum Weighing Designs

Abstract

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].