Privacy-Aware Data Integration for Enhanced Quantile Inference under Heterogeneity

Quantile estimation and inference play essential roles in diverse scientific and industrial applications, and their accuracy can often be enhanced by integrating auxiliary data from multiple sites. However, developing efficient aggregation methods for quantile inference under potential privacy constraints, particularly with heterogeneous datasets, remains challenging. To address these issues, we propose a systematic framework for quantile estimation and inference under potential local differential privacy (LDP). The key idea is to construct weighted estimators by adaptively aggregating quantile estimates from target and source sites. The adaptive weights are determined by minimizing the asymptotic variance, incorporating an additional $\ell_2$ penalty to account for parameter shift. A parallel stochastic gradient descent algorithm under LDP constraints is developed for weight estimation and valid inference. Additionally, we introduce a conservative weighted estimator to ensure robust inference across diverse heterogeneous scenarios. Rigorous theoretical analysis establishes the consistency, normality, and effectiveness of the proposed methods. Extensive numerical studies and real data application corroborate our theoretical findings.

Many decisions require estimating quantiles—cutoffs such as median income, low-income thresholds, or high-risk financial losses—not just averages. These estimates improve when data from several sites can be combined, but raw records may be private and different sites may describe different populations. This paper introduces a way to estimate a target quantile by safely borrowing information from other sites under local differential privacy, where each record is randomized before it leaves its site. The method estimates a quantile and its uncertainty at each site, then gives larger weights to sites that are more accurate and more similar to the target site. Noisier or more different sources receive smaller weights, so they are less likely to distort the final answer. The paper also provides a conservative version for cases where it is unclear whether another site is close enough to help. Theory shows that the uncertainty ranges are valid under stated conditions and can be shorter than those from using the target site alone when useful source data exist. Simulations and an income-data study show that the approach improves accuracy while respecting privacy constraints, which can help finance, health care, and other multi-site data settings.