Decentralized Locally Private Inference for Quantiles

We study quantile inference under local differential privacy for heterogeneous data streams distributed over a peer-to-peer network without a trusted central server. The proposed procedure targets a weighted mixture quantile using one-bit randomized-response scores and blockwise gossip, with each observation processed once and communication restricted to neighboring nodes. The main challenge is that incomplete synchronization creates persistent network disagreement that interacts with discontinuous quantile scores. We derive graph-dependent error bounds and establish asymptotic normality and a functional central limit theorem for the spatial–temporal average under suitable mixing and growth conditions. These results yield self-normalized confidence intervals without estimating the mixture density or the privacy-inflated score variance. Under stronger balance conditions, node-wise temporal averages inherit the same first-order limits, enabling inference from a single node's trajectory without additional network-wide aggregation. Numerical experiments and a real-data application examine the effects of privacy, communication frequency, and network connectivity on estimation and inference.