Density estimation under local differential privacy and Hellinger loss

In this paper, we carry out a piecewise constant estimator of the density for privatised data. We establish a non-asymptotic oracle inequality for the Hellinger loss and deduce that our estimator is adaptive and rate optimal over a wide range of Besov classes (up to possible logarithmic factors). We also get better estimation rates when the density is not only in a Besov class but also bounded away from 0. These rates are optimal within possible log factors. This result is in contrast to what happens with the L2 loss where the privatised minimax rates over Besov classes can be improved in some cases by assuming the target bounded from above.