Copula-based stratified randomized response for differential privacy: Laplace and Gaussian noise comparisons

Accurate estimation under privacy constraints is a critical challenge in modern data analysis, particularly when responses exhibit complex dependence structures and heavy-tailed noise. In this study, we investigate stratified randomized response mechanisms combined with differential privacy (DP), incorporating copula-based modeling to capture dependence between multiple noise components. We consider Laplace and Gaussian noise distributions and examine the effects of copula family, dependence parameter (θ), privacy level (ε), and stratum-specific randomization probabilities on bias and variance. Through extensive simulations, we demonstrate that Laplace noise amplifies variability and is highly sensitive to copula dependence, while Gaussian noise provides more stable estimates. Clayton and Gaussian copulas offer robust performance under moderate dependence, whereas Gumbel copulas exhibit extreme bias and variance under high dependence. Real data analysis using the mtcars dataset corroborates these findings, showing that copula-based modeling improves estimation stability compared to ignoring dependence. Our results provide practical guidance for selecting copula families, noise types, and privacy parameters, highlighting the trade-offs between privacy and statistical accuracy in privacy-preserving data collection.