2017/01/23 by Kazuto Fukuchi, Jun Sakuma, Fukuchi, Kazuto +1 · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning and Algorithms #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1701.06381
openalex publication_date 2017/01/23 · openalex created_date 2017/02/03 · openalex updated_date 2026/07/28
In this paper, we consider estimators for an additive functional of ϕ, which is defined as θ(P;ϕ)=∑i=1kϕ(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-4. Furthermore, if the divergence speed of the fourth derivative of ϕ is p4-α for α∈ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as \frack2(nln n)2α + \frack2-2αn.