Nonparametric density estimation for intentionally corrupted functional data

TL;DR

This paper develops a nonparametric method to estimate the density of original functional data from intentionally corrupted observations, ensuring privacy while maintaining statistical accuracy.

Contribution

It introduces a novel estimator for Wiener densities in privacy-preserving functional data models, with theoretical error bounds and data-driven parameter selection.

Findings

Estimator achieves near-optimal convergence rates.

Data-driven parameter choice does not significantly degrade performance.

Numerical experiments validate the method's effectiveness.

Abstract

We consider statistical models where functional data are artificially contaminated by independent Wiener processes in order to satisfy privacy constraints. We show that the corrupted observations have a Wiener density which determines the distribution of the original functional random variables, masked near the origin, uniquely, and we construct a nonparametric estimator of that density. We derive an upper bound for its mean integrated squared error which has a polynomial convergence rate, and we establish an asymptotic lower bound on the minimax convergence rates which is close to the rate attained by our estimator. Our estimator requires the choice of a basis and of two smoothing parameters. We propose data-driven ways of choosing them and prove that the asymptotic quality of our estimator is not significantly affected by the empirical parameter selection. We examine the numerical performance of our method via simulated examples.