  It is of soaring demand to develop statistical analysis tools that are robust
against contamination as well as preserving individual data owners' privacy. In
spite of the fact that both topics host a rich body of literature, to the best
of our knowledge, we are the first to systematically study the connections
between the optimality under Huber's contamination model and the local
differential privacy (LDP) constraints.
  In this paper, we start with a general minimax lower bound result, which
disentangles the costs of being robust against Huber's contamination and
preserving LDP. We further study four concrete examples: a two-point testing
problem, a potentially-diverging mean estimation problem, a nonparametric
density estimation problem and a univariate median estimation problem. For each
problem, we demonstrate procedures that are optimal in the presence of both
contamination and LDP constraints, comment on the connections with the
state-of-the-art methods that are only studied under either contamination or
privacy constraints, and unveil the connections between robustness and LDP via
partially answering whether LDP procedures are robust and whether robust
procedures can be efficiently privatised. Overall, our work showcases a
promising prospect of joint study for robustness and local differential
privacy.
