Minimax Optimal Additive Functional Estimation with Discrete Distribution
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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^kϕ(p_i) for a function ϕ, which covers the most of the entropy-like criteria. The minimax optimal risk of this problem has been already known for some specific ϕ, such as ϕ(p)=p^α and ϕ(p)=-p p. However, there is no generic methodology to derive the minimax optimal risk for the additive function estimation problem. In this paper, we reveal the property of ϕ that characterizes the minimax optimal risk of the additive functional estimation problem; this analysis is applicable to general ϕ. More precisely, we reveal that the minimax optimal risk of this problem is characterized by the divergence speed of the function ϕ.
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