Abstract: We consider statistical models in which the 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 that uniquely determines the distribution of the original functional random variables masked near the origin, and 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 establish an asymptotic lower bound on the minimax convergence rates that is close to the rate attained by our estimator. Our estimator requires choosing a basis and two smoothing parameters. We propose data-driven ways to do so and prove that the asymptotic quality of our estimator is not significantly affected by the empirical parameter selection. Lastly, we examine the numerical performance of our method using simulated examples.

Key words and phrases: Classification, convergence rates, differential privacy, infinite- dimensional Gaussian mixtures, Wiener densities.