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Our research delves into the balance between maintaining privacy andpreserving statistical accuracy when dealing with multivariate data that issubject to \textit{componentwise local differential privacy} (CLDP). With CLDP,each component of the private data is made public through a separate privacychannel. This allows for varying levels of privacy protection for differentcomponents or for the privatization of each component by different entities,each with their own distinct privacy policies. We develop general techniquesfor establishing minimax bounds that shed light on the statistical cost ofprivacy in this context, as a function of the privacy levels $\alpha_1, ... ,\alpha_d$ of the $d$ components. We demonstrate the versatility and efficiencyof these techniques by presenting various statistical applications.Specifically, we examine nonparametric density and covariance estimation underCLDP, providing upper and lower bounds that match up to constant factors, aswell as an associated data-driven adaptive procedure. Furthermore, we quantifythe probability of extracting sensitive information from one component byexploiting the fact that, on another component which may be correlated with thefirst, a smaller degree of privacy protection is guaranteed.

Minimax rate for multivariate data under componentwise local differential privacy constraints