Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions

In this paper, we consider estimators for an additive functional of $\varphi$, which is defined as $\theta (P;\varphi )={\sum }_{i=1}^k\varphi (p_i)$, from $n$ i.i.d. random samples drawn from a discrete distribution $P=(p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k)$ 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 $\varphi$ 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 $\varphi$ is larger than $p^{-4}$. Furthermore, if the divergence speed of the fourth derivative of $\varphi$ is $p^{4-\alpha }$ for $\alpha \in (0,1)$, the minimax optimal rate is obtained within a universal multiplicative constant as $\frac{k^2}{(n\ln n)^{2\alpha}} + \frac{k^{2-2\alpha}}{n}$.