In many statistical procedures, such as smoothing, resampling, or privacy-preserving release, it is convenient to transform the empirical distribution according to the task. We introduce an averaging transformation that assigns equal mass to $m$ estimated expected order statistics, where $m$ acts as a resolution parameter. This yields finite-sample properties and a tractable route to asymptotic analysis. The distance between the transformed empirical distribution and its deterministic counterpart is bounded by the estimation error of the classical empirical distribution, in both Wasserstein and integral metrics. Moreover, every $L$-functional of the transformed distribution corresponds to an $L$-functional of the classical one, with updated weights. We establish almost sure and weak convergence, asymptotic distributions for distance-based functionals, and bootstrap validity. Finally, we obtain a differentially private release with explicit accuracy bounds, and compare it with private histograms in simulations and data.

翻译：暂无翻译