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

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