A Probability Proportional to Size Estimation of a Rare Sensitive Attribute Using a Partial Randomized Response Model with Poisson Distribution

In this paper, we suggest using a partial randomized response model using Poisson distribution to efficiently estimate a rare sensitive attribute by applying the probability proportional to size (PPS) sampling method when the population is composed of several different and sensitive clusters. We have obtained estimators for a rare and sensitive attribute and their variances and variance estimates by applying PPS sampling and two-stage equal probability sampling. We compare the efficiency between the estimators of the rare sensitive attribute, one obtained via PPS sampling with replacement and the other obtained using the two-stage equal probability sampling with replacement. As a result, it is confirmed that the estimate obtained via the PPS sampling with replacement is more efficient than the estimate provided by the two-stage equal probability sampling with replacement when the cluster sizes are different.