Randomized Fractional Weighing Designs

Abstract

Consider the problem of estimating unbiasedly a linear function, $\lambda?' \omega, \omega' = (\omega_1, \cdots, \omega_p)$, of $p$ weights $\omega_1, \cdots, \omega_p (0 < \omega_i < \infty$ for all $i = 1, \cdots, p)$ when the number of possible weighing operations, $n$, is smaller than $p$. Assume further that the weighing design is of the chemical type (each entry of the design matrix can assume one of the values $-1, 0, +1)$. A weighing design in which the number of weighing operations is smaller than the number of objects, $n < p$, will be called a fractional weighing design. A fractional weighing design is singular. As is well known, if a non-randomized weighing design is singular, one cannot estimate unbiasedly each of the $p$ weights. Some linear functions $\lambda?'\omega$ can be estimated, however, unbiasedly. For example, the sum of all $p$ weights can be estimated unbiasedly by a singular weighing design in which, at each weighing operation, all the objects are placed on one pan. Such a singular weighing design is in fact optimal. One can not attain the same precision in estimating the sum of all the $p$ weights by any non-singular design (see K. S. Banerjee [1]). Some linear functions of the weights can thus be estimated, with a sufficient degree of precision, on the basis of a number of weighing operations, $n$, smaller than the number of weights, $p$. A. M. Mood [4] has shown that if the design matrix of a non-singular weighing design is a Hadamard matrix (if it exists) then the design is optimal. We shall therefore study fractional weighing designs based on Hadamard matrices. If $p$ is such that no Hadamard matrix of order $p$ exists, we shall consider the smallest integer $p'$ larger than $p$, for which a Hadamard matrix exists and add $(p' - p)$ dummy weights. The linear functions $\lambda?'\omega$ can be extended in a way that assigns the dummy weights the coefficient zero. We shall then consider randomized fractional weighing designs which are constructed by choosing at random $n$ rows, independently and with replacement, from the given Hadamard matrix according to a probability vector $\xi$ of order $p$. Every such row specifies a weighing operation to be performed. Non-randomized fractional designs are special cases of the randomized designs under consideration. It is well known that if the weighing design is fractional $(n < p)$ then not all the linear functions $\lambda?'\omega$ are estimable under non-randomized designs. It is shown in the present paper that any linear function $\lambda?'\omega$ can be estimated unbiasedly under a proper randomization scheme. Optimal randomization procedures depend on the functional $\lambda?$. Complete class of unbiased estimation procedures is given for any linear function $\lambda?'\omega$. It is shown that every functional $\lambda?$ specifies a subset of, say $r (1 \leqq r \leqq p)$, admissible weighing operations (rows of the Hadamard matrix); in the sense that if other weighing operations are chosen then, either the estimation procedure is biased, or has a variance larger than the one that can be attained under the admissible weighing operations. It is also proven that if each of the $r$ admissible weighing operations $(1 \leqq r \leqq p)$ is chosen with probability equal to $1/r$, then the corresponding unbiased estimator has a uniformly (in $\omega$ and $\sigma^2$) minimum variance, relative to all the above specified randomization procedures. The formulae of the unbiased estimator of $\lambda?'\omega$ and its variance are extended in Section 4 to the case of random choice without replacement, of $n$ rows out of the $r$ admissible ones. This randomization procedure improves the results that can be obtained under the best randomization procedure with replacement. Finally, the relationship of the present study to the studies of S. Ehrenfeld and S. Zacks [2], [3] is discussed in Section 5.