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

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This paper addresses an estimation problem of an additive functional of ϕ, which is defined as θ(P;ϕ)=∑ᵢ₌₁ᵏϕ(pᵢ), given n i.i.d. random samples drawn from a discrete distribution P=(p₁,...,pₖ) 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 ϕ 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 α of the divergence speed. For α∈ (1,3/2), we show that the minimax rate is 1/n+k²(n n)²α. Besides, we show that the minimax rate is 1/n for α∈ [3/2,2].