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

翻译：我们的研究深入探讨了在受分量级局部差分约束的多元数据中，隐私保护与统计精度之间的权衡。在分量级局部差分隐私框架下，私有数据的每个分量通过独立的隐私信道公开，从而允许为不同分量设置差异化的隐私保护强度，或由不同实体依据各自隐私策略对每个分量进行私有化处理。我们开发了建立极小化界限的通用技术，以揭示该背景下隐私的统计代价——体现为$d$个分量的隐私参数$\alpha_1,...,\alpha_d$的函数。通过展示多项统计应用，我们证明了这些技术的普适性与高效性。具体而言，我们研究了分量级局部差分隐私下的非参数密度估计与协方差估计，提供了常数因子匹配的上下界及相应的数据驱动自适应程序。此外，我们量化了通过利用某分量（该分量可能与另一具有较低隐私保护强度的分量存在相关性）窃取敏感信息的概率。