  In this work, we study local minimax convergence estimation rates subject to
$\epsilon$-differential privacy. Unlike worst-case rates, which may be
conservative, algorithms that are locally minimax optimal must adapt to easy
instances of the problem. We construct locally minimax differentially private
estimators for one-parameter exponential families and estimating the tail rate
of a distribution. In these cases, we show that optimal algorithms for simple
hypothesis testing, namely the recent optimal private testers of Canonne et al.
(2019), directly inform the design of locally minimax estimation algorithms.
