### Abstract:

Two topics concerning statistical sampling of initiative petitions are considered in this dissertation. The first concerns statistical estimation of the number of distinct valid signatures in a petition, and the second evaluates the statistical decision rule used by Oregon for determining certification of state initiative and referendum petitions.
In several states that permit initiative petitions to modify or add legislation, statistical sampling of signatures is used to obtain an estimate for the number of distinct valid signatures in the petitions. This estimate depends on the number of signatures submitted, the number of invalid signatures and, in some states, the number of duplicates of valid signatures.
We consider several linear estimators and a non-linear estimator for the number of distinct valid signatures. Their performances are compared with respect to bias and root mean squared error using several sample sizes for four fully-verified petitions from Washington State. Exact expressions for the bias and root mean square error are used for the linear estimators and estimates from simulated random sampling are used for the nonlinear estimators. For the small sampling fractions typically used in state initiative petitions (3-10%), none of the estimators are found to perform much better than the estimator that is constructed to be unbiased when valid signatures are assumed to be duplicated at most once.
Oregon allows a petition to be filed in either one or two submissions. The Oregon decision rule for certification of petitions is complicated in that multiple stages of sampling are used: two stages for a single submission and three stages for two submissions. A petition can be accepted (certified) after any sampling stage but only rejected after verifying all samples from each submission. The decision rule is based on estimates for the number of distinct valid signatures obtained at each sampling stage. We evaluate the performance of the Oregon decision rule by calculating an approximation for the probability of making a correct decision for the certification of several hypothetical petitions. The petitions are chosen to represent different sizes and quality with respect to invalid and duplicated valid signatures.