  We revisit the classical problem of nonparametric density estimation but
impose local differential privacy constraints. Under such constraints, the
original multivariate data $X_1,\ldots,X_n \in \mathbb{R}^d$ 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. We illustrate the trade-off between data utility and
privacy by means of a small simulation study.
