Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions [article]

Kazuto Fukuchi, Jun Sakuma
2017 arXiv   pre-print
In this paper, we consider estimators for an additive functional of ϕ, which is defined as θ(P;ϕ)=∑_i=1^kϕ(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
more » ... imator 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∈ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as k^2/(n n)^2α + k^2-2α/n.
arXiv:1701.06381v3 fatcat:bcddnbm55jfkdk4i3o5shxrlcy