Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions
Minimax Optimal Estimators for Additive Scalar Functionals of Discrete Distributions
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离散分布的加性标量泛函的极小极大最优估计
DOI：
10.1109/isit.2017.8006900
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Kazuto Fukuchi and Jun Sakuma
中科院分区：
文献类型：
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作者：
石川裕卓;佐藤博隆;加美山隆;Kazuto Fukuchi and Jun Sakuma;Kazuto Fukuchi and Jun Sakuma
本文考虑φ的一个加性泛函的估计量，它定义为θ(P；φ)=ΣKi=1φ(Pi))，来自nI.D.从字母表大小为k的离散分布P=(p1，…，pk)中随机抽取样本。对于加性泛函的估计问题，我们提出了一个极小极大最优估计。我们发现，当φ的四阶导数的发散速度较高时，极小极大最优速率是由发散速度表征的。结果表明，当φ的四阶导数的发散速度大于p-4时，不存在一致估计。此外，如果φ的四阶导数的发散速度对α(0，1)是p4-αϵ，则在一个普适乘法常数K2/(N Ln N)2α+K2-2α/n内可得到极小极大最优速率。
In this paper, we consider estimators for an additive functional of φ, which is defined as θ(P; φ) = Σki=1φ(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-αfor α ϵ (0,1), the minimax optimal rate is obtained within a universal multiplicative constant as k2/(n ln n)2α+ k2-2α/n.