Sensitive data must be protected and randomized before their public release. Local differential privacy (LDP) is the most common setup to quantify the amount of privacy in the released data. The challenge is then to make inference on the true underlying population using the private sample only. Usually, a trade-off takes place between the amount of privacy introduced and the quality of the statistical inference.
Nonparametric estimation of functions and functionals has been proven slower in the rate under LDP when compared to the non-private setup. It was noticed that both the privacy mechanism and the statistical procedures depend on the smoothness of the underlying probability density. We will give here new privacy mechanisms and associated estimators of the quadratic functional that are free of the smoothness parameter.
