  The rate-privacy function is defined in \cite{Asoodeh} as a tradeoff between
privacy and utility in a distributed private data system in which both privacy
and utility are measured using mutual information. Here, we use maximal
correlation in lieu of mutual information in the privacy constraint. We first
obtain some general properties and bounds for maximal correlation and then
modify the rate-privacy function to account for the privacy-constrained
estimation problem. We find a bound for the utility in this problem when the
maximal correlation privacy is set to some threshold $\epsilon>0$ and construct
an explicit privacy scheme which achieves this bound.
