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.