Multivariate density estimation from privatised data: universal consistency and minimax rates

We revisit the classical problem of nonparametric density estimation but impose local differential privacy constraints. Under such constraints, the original multivariate data $ X_1,\ldots,X_n \in \mathbb {R}^d $ X1,…,Xn∈Rd cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of an observed vector. Based on these randomised data, we propose a novel estimator of the density function, which can be viewed as a privatised version of the well-studied histogram density estimator. Our theoretical results include universal pointwise consistency and strong universal $ L_1 $ L1-consistency. In addition, a convergence rate for Lipschitz continuous functions is derived, which is complemented by a matching minimax lower bound. We illustrate the trade-off between data utility and privacy by means of a small simulation study.