论文标题
在当地差异隐私下的自适应点密度估计
Adaptive pointwise density estimation under local differential privacy
论文作者
论文摘要
我们考虑在固定点处的密度估计在局部差异隐私约束下的估计，在此观察结果是在可用于统计推断之前匿名的。我们提出了投影密度估计器的私有化版本以及内核密度估计器，并在隐私约束下得出其最小值。由于匿名化，最小值速率有双重恶化，我们证明这是不可避免的，可以通过提供下限。在这两个估计过程中，都必须选择一个调谐参数。我们建议分别选择带宽和截止维度的经典Goldenshluger-Lepski方法的一种变体，并分析其性能。它提供自适应的最小值 - 最佳（最大对数因子）估计器。我们详细讨论了上限和上限如何取决于隐私约束，而自适应方法的修改反映了这种约束。
We consider the estimation of a density at a fixed point under a local differential privacy constraint, where the observations are anonymised before being available for statistical inference. We propose both a privatised version of a projection density estimator as well as a kernel density estimator and derive their minimax rates under a privacy constraint. There is a twofold deterioration of the minimax rates due to the anonymisation, which we show to be unavoidable by providing lower bounds. In both estimation procedures a tuning parameter has to be chosen. We suggest a variant of the classical Goldenshluger-Lepski method for choosing the bandwidth and the cut-off dimension, respectively, and analyse its performance. It provides adaptive minimax-optimal (up to log-factors) estimators. We discuss in detail how the lower and upper bound depend on the privacy constraints, which in turn is reflected by a modification of the adaptive method.
