The sampling distribution of the W estimator of the number of valid signatures on a petition

Abstract

A new non linear estimator, W, for the number of valid, unique signatures on a petition has been shown better, for the cases enumerated and with certain restrictions, than a popular Goodman-type statistic, G. This article extends those results with relaxed conditions by developing the exact probability mass function and mean of W and a close approximation of the variance (Var(W)). If the proportion of valid signatures among unique and duplicated signatures is the same, then Var(W) is approximately a function of the means and variances of the two sample statistics. Using the delta method, we estimate Var(W), with the resulting approximation shown to be good, even when the condition of equal proportions does not hold. We compare W to G and establish which estimator is preferred for different intervals of the design parameters. Data from a Washington State petition illustrate the findings.