  We consider the minimax estimation problem of a discrete distribution with
support size $k$ under locally differential 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.
  In our previous work (IEEE Trans. Inform. Theory, 2018), we proposed a family
of new privatization schemes and the corresponding estimator. We also proved
that our scheme and estimator are order optimal in the regime $e^{\epsilon} \ll
k$ under both $\ell_2^2$ (mean square) and $\ell_1$ loss. In this paper, we
sharpen this result by showing asymptotic optimality of the proposed scheme
under the $\ell_p^p$ loss for all $1\le p\le 2.$ More precisely, we show that
for any $p\in[1,2]$ and any $k$ and $\epsilon,$ the ratio between the
worst-case $\ell_p^p$ estimation loss of our scheme and the optimal value
approaches $1$ as the number of samples tends to infinity. The lower bound on
the minimax risk of private estimation that we establish as a part of the proof
is valid for any loss function $\ell_p^p, p\ge 1.$
