$L_1$ density estimation from privatised data

We revisit the classical problem of nonparametric density estimation, but impose local differential privacy constraints. Under such constraints, the original data $X_1,\ldots,X_n$, taking values in $\mathbb{R}^d$, 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 a vector $X_i$. Based on these randomised data, we design 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 $L_1$-consistency. In addition, a convergence rate over classes of Lipschitz functions is derived, which is complemented by a matching minimax lower bound.