  Differential privacy is a cryptographically-motivated approach to privacy
that has become a very active field of research over the last decade in
theoretical computer science and machine learning. In this paradigm one assumes
there is a trusted curator who holds the data of individuals in a database and
the goal of privacy is to simultaneously protect individual data while allowing
the release of global characteristics of the database. In this setting we
introduce a general framework for parametric inference with differential
privacy guarantees. We first obtain differentially private estimators based on
bounded influence M-estimators by leveraging their gross-error sensitivity in
the calibration of a noise term added to them in order to ensure privacy. We
then show how a similar construction can also be applied to construct
differentially private test statistics analogous to the Wald, score and
likelihood ratio tests. We provide statistical guarantees for all our proposals
via an asymptotic analysis. An interesting consequence of our results is to
further clarify the connection between differential privacy and robust
statistics. In particular, we demonstrate that differential privacy is a weaker
stability requirement than infinitesimal robustness, and show that robust
M-estimators can be easily randomized in order to guarantee both differential
privacy and robustness towards the presence of contaminated data. We illustrate
our results both on simulated and real data.
