On Non-randomized Fractional Weighing Designs

Abstract

It is known that, for the estimation of $p$ individual weights, the optimum weighing design [3] for a chemical balance is given [4] by a Hadamard matrix $X$ of dimensions $p \times p$, when it exists. If $r$ rows of $X$ are used for the weighing operations, the resultant design matrix $X_1$ of dimensions $r \times p$ will be a fraction of the full design matrix $X$, and will necessarily be singular. While it is not possible, with such a fractional weighing design, to furnish unique and unbiased estimates of the individual weights, it may be practicable, however, to afford a unique and unbiased estimate of a linear function of the weights. In a recent paper, Zacks [6] has considered questions of admissibility of "randomization procedures" for such fractional weighing designs and has indicated a few basic results in this direction proceeding on the same lines as followed in [2]. We furnish, in this paper, some results of connected interest with respect to such fractional weighing designs without resorting to any randomization procedure in the selection of rows of the full design matrix $X$. Apart from this connection, the results are expected to have an importance of their own. We have spelled out here the structure of the estimate of the estimable linear function along with its variance, bringing out the connection of this variance with the variance as obtainable with the full design matrix. And, in the process, it has been indicated, in relation to the fraction used, to what extent we can afford to be arbitrary in the selection of the components of $\lambda_p$ which enters into the estimable linear function $\lambda_p'\beta_p$, where $\lambda_p$ and $\beta_p$ are $p \times 1$ column vectors representing the coefficients and the weights respectively. We have shown that, depending on $\lambda_p$, we can obtain, with a fraction, the same precision for the estimate without having to perform all the weighing operations as required in a full design matrix. In such situations, repetitions of the fraction would lead to increased precision as compared to the adoption of the full design matrix.