Minimax Optimal Additive Functional Estimation with Discrete Distribution: Slow Divergence Speed Case

This paper addresses an estimation problem of an additive functional of $\varphi$, which is defined as $\theta (P;\varphi )={\sum }_{i=1}^k\varphi (p_i)$, given $n$ i.i.d. random samples drawn from a discrete distribution $P=(p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k)$ with alphabet size $k$. We have revealed in the previous paper that the minimax optimal rate of this problem is characterized by the divergence speed of the fourth derivative of $\varphi$ in a range of fast divergence speed. In this paper, we prove this fact for a more general range of the divergence speed. As a result, we show the minimax optimal rate of the additive functional estimation for each range of the parameter $\alpha$ of the divergence speed. For $\alpha \in (1,3/2)$,weshowthattheminimaxrateis$\frac{1}{n}+\frac{k^2}{(n\ln n)^{2\alpha}}$\mathrm{.}Besides,weshowthattheminimaxrateis$\frac{1}{n}$for$\alpha \in [3/2,2]$.