  We consider the estimation of a density at a fixed point under a local
differential privacy constraint, where the observations are anonymised before
being available for statistical inference. We propose both a privatised version
of a projection density estimator as well as a kernel density estimator and
derive their minimax rates under a privacy constraint. There is a twofold
deterioration of the minimax rates due to the anonymisation, which we show to
be unavoidable by providing lower bounds. In both estimation procedures a
tuning parameter has to be chosen. We suggest a variant of the classical
Goldenshluger-Lepski method for choosing the bandwidth and the cut-off
dimension, respectively, and analyse its performance. It provides adaptive
minimax-optimal (up to log-factors) estimators. We discuss in detail how the
lower and upper bound depend on the privacy constraints, which in turn is
reflected by a modification of the adaptive method.
