Minimax optimal estimators for additive scalar functionals of discrete distributions

摘要

In this paper, we consider estimators for an additive functional of Φ, which is defined as θ(P; Φ) = Σi=1 Φ(pi), from n i.i.d. random samples drawn from a discrete distribution P = (p1,…, pk) with alphabet size k. We propose a minimax optimal estimator for the estimation problem of the additive functional. We reveal that the minimax optimal rate is characterized by the divergence speed of the fourth derivative of Φ if the divergence speed is high. As a result, we show there is no consistent estimator if the divergence speed of the fourth derivative of Φ is larger than p. Furthermore, if the divergence speed of the fourth derivative of Φ is p for α ∊ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as k/(n ln n) + k.