Locally Differentially Private Online Quantile Inference with an Adaptive Mechanism

A local privacy mechanism chosen for frequency estimation need not produce an efficient joint confidence set for quantiles. We select the reporting channel for ellipsoid volume or maximum simultaneous interval width, using one locally private categorical report per user. At the target quantiles, category probabilities are determined by the requested levels, making each candidate's reference score law public. Unknown quantile densities cancel from the volume comparison; an independent private probe supplies them for interval-width selection. Under local smoothness and bounded-library conditions, we establish exact library minimization for volume, a plug-in oracle inequality for width, and conditional asymptotic coverage after selection. A fixed-threshold correction separates localization error from the final score average. Independent strong-privacy experiments give selected joint coverage of $0.93$–$0.95$ and 42%–43% smaller ellipsoid volumes than randomized response under the same correction, including the users reserved for localization. The comparison connects mechanism choice to confidence geometry and quantifies the precision cost of localization.