Minimax optimal estimators for additive scalar functionals of discrete distributions

In this paper, we consider estimators for an additive functional of φ, which is defined as θ(P; φ) = Σ
^k
_i=1
φ(p
_i
), from n i.i.d. random samples drawn from a discrete distribution P = (p
₁
,..., p
_k
) 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
^-4
. Furthermore, if the divergence speed of the fourth derivative of φ is p
^4-α
for α ϵ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as k
²
/(n ln n)
^2α
+ k
^2-2α
/n.