  We examine privacy-preserving inferences of group mean differences in
zero-inflated right-skewed (zirs) data. Zero inflation and right skewness are
typical characteristics of ads clicks and purchases data collected from
e-commerce and social media platforms, where we also want to preserve user
privacy to ensure that individual data is protected. In this work, we develop
likelihood-based and model-free approaches to analyzing zirs data with formal
privacy guarantees. We first apply partitioning and censoring (PAC) to
``regularize'' zirs data to get the PAC data. We expect inferences based on PAC
to have better inferential properties and more robust privacy considerations
compared to analyzing the raw data directly. We conduct theoretical analysis to
establish the MSE consistency of the privacy-preserving estimators from the
proposed approaches based on the PAC data and examine the rate of convergence
in the number of partitions and privacy loss parameters. The theoretical
results also suggest that it is the sampling error of PAC data rather than the
sanitization error that is the limiting factor in the convergence rate. We
conduct extensive simulation studies to compare the inferential utility of the
proposed approach for different types of zirs data, sample size and partition
size combinations, censoring scenarios, mean differences, privacy budgets, and
privacy loss composition schemes. We also apply the methods to obtain
privacy-preserving inference for the group mean difference in a real digital
ads click-through data set. Based on the theoretical and empirical results, we
make recommendations regarding the usage of these methods in practice.
