Singularity in Hotelling's Weighing Designs and a Generalized Inverse

Abstract

Results of $N$ weighing operations to determine the individual weights of $p$ objects, as envisaged in Hotelling's weighing designs [4], fit into the linear model $Y = X\beta + e$, where $Y$ is an $N \times 1$ random observed vector of the recorded results of weighings; $X = (x_{ij}), i = 1, 2, \cdots, N, j = 1, 2, \cdots, p$, is an $N \times p$ matrix of known quantities with $x_{ij} = +1, -1$ or 0, if, in the $i$th weighing operation, the $j$th object is placed respectively in the left pan, right pan, or in none; $\beta$ is a $p \times 1$ vector $(p \leqq N)$ representing the weights of the objects; $e$ is an $N \times 1$ unobserved random vector such that $E(e) = 0$ and $E(ee') = \sigma^2I_N. X$ represents the weighing design matrix. When $X$ is of full rank, that is, when $X'X$ is non-singular, the weights of the objects are given by the least squares estimates, $\hat\beta = \lbrack X'X\rbrack^{-1}X'Y$. The covariance matrix is given by $\operatorname{Cov}(\hat\beta) = \sigma^2\lbrack X'X\rbrack^{-1} = \sigma^2C. c_{ii}$, which is the $i$th diagonal element of $C$, represents the variance factor for the $i$th object. In weighing designs, we search for the elements $x_{ij}$ such that $c_{ii}$ is the least for each $i$. When, however, $X$ is singular, it is well known that, while it is not possible to have a unique (unbiased) estimate for each of the $p$ objects, it is possible to have a unique (unbiased) estimate for a linear function, $\lambda'\beta$, of the parameters, if and only if there exists a solution for $r$ in the equations $Sr = \lambda$, where $S = X'X$. Raghavarao [7] visualized that bad designing, repetitions or accidents might lead to singular weighing designs, and considered the question of taking additional weighings to make the resultant design matrix $X$ of full rank maximizing the resultant $\det. |X'X|$, if possible, as required in Mood's [5] efficiency definition. He appeared to take only the chemical balance (and not the spring balance) into consideration, and was eventually led to the question of dealing with the situation when the rank of $X$ was less than the full by only one. In this paper, the problem is, in the first place, cast into the framework of a generalized inverse (referred to as a $g$-inverse). Two harmonizing results in the area of $g$-inverses are then indicated by way of an aid to algebraic simplifications in the context of tackling the general problem. Singular weighing designs may not perhaps all be altogether useless in themselves, although one may not adopt singular weighing designs in a scheme of weighing operations to begin with. In certain given situations, singular weighing designs may even be preferred to the best available weighing designs adopted for the estimation of the weight of each individual object, and, in such situations, there may arise a necessity for comparing two singular weighing designs. Linking the problem of singular weighing designs to a $g$-inverse, as has been done in this paper, would also help institute such a comparison. Consideration is further given to the situation when the rank of $X$ may be less than the full by more than one. Finally, a special class of singular spring balance designs is also discussed.