On consistent and rate optimal estimation of the missing mass

IF 1.5 Q2 PHYSICS, MATHEMATICAL
Fadhel Ayed, M. Battiston, F. Camerlenghi, S. Favaro
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引用次数: 8

Abstract

. Given n samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the ( n + 1)-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results have shown: i) the impossibility of estimating the missing mass without imposing further assumptions on type’s proportions; ii) the consistency of the Good-Turing estimator of the missing mass under the assumption that the tail of type’s proportions decays to zero as a regularly varying function with parameter α ∈ (0 , 1); ii) the rate of convergence n − α/ 2 for the Good-Turing estimator under the class of α ∈ (0 , 1) regularly varying P . In this paper we introduce an alternative, and remarkably shorter, proof of the impossibility of a distribution-free estimation of the missing mass. Beside being of independent interest, our alternative proof suggests a natural approach to strengthen, and expand, the recent results on the rate of convergence of the Good-Turing estimator under α ∈ (0 , 1) regularly varying type’s proportions. In particular, we show that the convergence rate n − α/ 2 is the best rate that any estimator can achieve, up to a slowly varying function. Furthermore, we prove that a lower bound to the minimax estimation risk must scale at least as n − α/ 2 , which leads to conjecture that the Good-Turing estimator is a rate optimal minimax estimator under regularly varying type proportions.
缺失质量的一致性和速率最优估计
. 给定n个样本,这些样本属于不同类型的个体,比例未知,我们如何估计在第(n + 1)次抽取时发现新类型的概率?这是统计学中的一个经典问题,通常被称为缺失质量估计问题。最近的结果表明:i)如果不对类型的比例施加进一步的假设,估计缺失质量是不可能的;ii)假设类型比例的尾部作为参数α∈(0,1)的正则变函数衰减为零,缺失质量的Good-Turing估计量的相合性;ii)在α∈(0,1)有规则变化P的类下,Good-Turing估计量的收敛速率n−α/ 2。在本文中,我们引入了另一种证明,而且非常简短,证明了不可能对缺失质量进行无分布估计。除了具有独立的兴趣之外,我们的替代证明提出了一种自然的方法来加强和扩展最近关于α∈(0,1)规则变化类型比例下Good-Turing估计的收敛率的结果。特别是,我们证明了收敛速率n−α/ 2是任何估计器可以达到的最佳速率,直到缓慢变化的函数。此外,我们证明了极大极小估计风险的下界必须至少缩放为n - α/ 2,从而推测在规则变化的类型比例下,Good-Turing估计量是一个速率最优极大极小估计量。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
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