Based on binary inquiries, we developed an algorithm to estimate population quantiles under Local Differential Privacy (LDP). By self-normalizing, our algorithm provides asymptotically normal estimation with valid inference, resulting in tight confidence intervals without the need for nuisance parameters to be estimated. Our proposed method can be conducted fully online, leading to high computational efficiency and minimal storage requirements with $\mathcal{O}(1)$ space. We also proved an optimality result by an elegant application of one central limit theorem of Gaussian Differential Privacy (GDP) when targeting the frequently encountered median estimation problem. With mathematical proof and extensive numerical testing, we demonstrate the validity of our algorithm both theoretically and experimentally.

翻译：基于二元查询，我们提出了一种在局部差分隐私（LDP）框架下估计总体分位数的算法。通过自标准化技术，该算法可生成渐进正态的估计结果并实现有效推断，从而在无需估计冗余参数的前提下获得紧凑置信区间。所提方法完全支持在线计算模式，具有极高的计算效率与仅需$\mathcal{O}(1)$空间的极低存储需求。针对常见的中位数估计问题，我们通过精巧运用高斯差分隐私（GDP）的中心极限定理，证明了该算法的最优性。通过理论推导与大规模数值实验，我们验证了该算法在理论与实证层面的有效性。