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.

中文翻译：

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故意损坏的功能数据的非参数密度估计

我们考虑统计模型，其中功能数据被独立的维纳过程人为污染，以满足隐私约束。我们表明，损坏的观测值具有一个维纳密度，它决定了原始函数随机变量的分布，在原点附近被唯一地屏蔽，并且我们构建了该密度的非参数估计量。我们推导出其平均积分平方误差的上限，该上限具有多项式收敛速度，并且我们建立了极小极大收敛速度的渐近下限，该下限接近我们的估计器获得的速度。我们的估计器需要选择一个基和两个平滑参数。我们提出了数据驱动的选择方法，并证明我们的估计器的渐近质量不受经验参数选择的显着影响。我们通过模拟示例检查我们方法的数值性能。