  We consider the minimax estimation problem of a discrete distribution with
support size $k$ under privacy constraints. A privatization scheme is applied
to each raw sample independently, and we need to estimate the distribution of
the raw samples from the privatized samples. A positive number $\epsilon$
measures the privacy level of a privatization scheme. For a given $\epsilon,$
we consider the problem of constructing optimal privatization schemes with
$\epsilon$-privacy level, i.e., schemes that minimize the expected estimation
loss for the worst-case distribution. Two schemes in the literature provide
order optimal performance in the high privacy regime where $\epsilon$ is very
close to $0,$ and in the low privacy regime where $e^{\epsilon}\approx k,$
respectively.
  In this paper, we propose a new family of schemes which substantially improve
the performance of the existing schemes in the medium privacy regime when $1\ll
e^{\epsilon} \ll k.$ More concretely, we prove that when $3.8 < \epsilon
<\ln(k/9) ,$ our schemes reduce the expected estimation loss by $50\%$ under
$\ell_2^2$ metric and by $30\%$ under $\ell_1$ metric over the existing
schemes. We also prove a lower bound for the region $e^{\epsilon} \ll k,$ which
implies that our schemes are order optimal in this regime.
