Differentially Private Quantile Inference: Phase Transitions and a Gaussianization Mechanism

Private quantile inference requires the distribution of sampling error together with the randomization used for privacy. We characterize this distribution for the rank exponential mechanism under local density regularity and a public bounded output interval. As ${ϵ}_n\to 0$ with $n{ϵ}_n\to \infty$, averaging its length-weighted order-statistic gaps yields a conditional Laplace limit independent of the sample fluctuation. This identifies the Gaussian, Gaussian–Laplace, and Laplace regimes, with privacy scale $1/(n{ϵ}_n)$ and no logarithmic factor. Neighboring private quantiles estimate the common local scale, giving feasible intervals across regimes. We then change the error distribution through a one-step correction with Gaussian noise added to a bounded quantile score. Under explicit pilot and nuisance conditions, this construction under approximate differential privacy has a Gaussian pivot at every relative noise level. Simulations illustrate the phase laws and show how public range and budget allocation determine the width of feasible rank-based intervals, including the tradeoff between center accuracy and scale estimation.