Estimation of a rare sensitive attribute in probability proportional to size measures using Poisson distribution

We propose new variants of Land et al.'s [Estimation of a rare sensitive attribute using Poisson distribution. Statistics. 2011. DOI: 10.1080/02331888.2010.524300] randomized response model when a population consists of some clusters and the population is stratified with some clusters in each stratum. The estimator for the mean number of persons who possess a rare sensitive attribute, its variance, and the variance estimator are suggested when the parameter of a rare unrelated attribute is assumed to be known and unknown. The clusters are selected with and without replacement. When they are selected with replacement, the selecting probabilities for each cluster are defined depending on the cluster sizes and with equal probability. In addition, the variance comparison between a probability proportional to size (PPS) and PPS for stratification are performed. When the parameters vary in clusters, the stratified PPS has better efficiency than the PPS.