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

Abstract

This paper addresses a problem of estimating an additive functional given n i.i.d. samples drawn from a discrete distribution P = (p1, ..., p k ) with alphabet size k. The additive functional is defined as θ(P ; φ) = k i=1 φ(pi) 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 ln 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.