This paper addresses a problem of estimating an additive functional given n i.i.d. samples drawn from a discrete distribution P = (p_1, ..., p_k) with alphabet size k. The additive functional is defined as θ(P; Φ) = ∑_(i=1)~k Φ(p_i) for a function Φ, which covers the most of the entropy-like criteria. We revealed in the previous paper [1] that the minimax optimal rate of this problem is characterized by the divergence speed, whereas the characterization is valid only when α ∈ (0, 1) where α denotes the parameter of the divergence speed. In this paper, we extend this characterization to a more general range of the divergence speed, including α ∈ (1,3/2) and α ∈ [3/2, 2]. As a result, we show that the minimax rates for α ∈ (1,3/2) and α ∈ [3/2,2] are 1/n + k~2/(n ln n)~(2α) and 1/n, respectively.
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