Title: Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions,
Abstract: In this paper, we consider estimators for an additive functional of $\phi$,
which is defined as $\theta(P;\phi)=\sum_{i=1}^k\phi(p_i)$, from $n$ i.i.d.
random samples drawn from a discrete distribution $P=(p_1,...,p_k)$ with
alphabet size $k$. We propose a minimax optimal estimator for the estimation
problem of the additive functional. We reveal that the minimax optimal rate is
characterized by the divergence speed of the fourth derivative of $\phi$ if the
divergence speed is high. As a result, we show there is no consistent estimator
if the divergence speed of the fourth derivative of $\phi$ is larger than
$p^{-4}$. Furthermore, if the divergence speed of the fourth derivative of
$\phi$ is $p^{4-\alpha}$ for $\alpha \in (0,1)$, the minimax optimal rate is
obtained within a universal multiplicative constant as $\frac{k^2}{(n\ln
n)^{2\alpha}} + \frac{k^{2-2\alpha}}{n}$.