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

This paper addresses an estimation problem of an additive functional of φ, which is defined as θ(P ;φ) = ∑ k i=1 φ(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 [1] 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 + k 2 (n lnn)2α . Besides, we show that the minimax rate is 1 n for α ∈ [3/2, 2].