On the non‐asymptotic and sharp lower tail bounds of random variables
On the non‐asymptotic and sharp lower tail bounds of random variables
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DOI：
10.1002/sta4.314
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发表时间：
2018-10
期刊：
影响因子：
1.7
通讯作者：
Anru R. Zhang;Yuchen Zhou
中科院分区：
文献类型：
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作者：
Anru R. Zhang;Yuchen Zhou
随机变量的非渐近尾界在概率、统计和机器学习中起着至关重要的作用。尽管文献中对尾概率上界的研究取得了很大的成功，但尾概率下界的研究却相对较少。在本文中，我们引入了系统的和用户友好的方案来开发尾部概率的非渐近下界。此外，我们为独立的亚高斯型和亚指数型随机变量的和建立了尖锐的下尾界，它们分别符合经典的Hoeffding型和Bernstein型浓度不等式。我们还提供了一组分布的非渐近匹配上下尾界，包括gamma， beta,（规则，加权和非中心）卡方，二项，泊松，Irwin-Hall等。我们应用这一结果建立了独立的次高斯和次指数随机变量和的极值期望的匹配上界和下界。最后考虑了稀疏非均质混合信号识别的统计应用。
The non‐asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probabilities in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user‐friendly schemes for developing non‐asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub‐Gaussian and sub‐exponential random variables, which match the classic Hoeffding‐type and Bernstein‐type concentration inequalities, respectively. We also provide non‐asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi‐square, binomial, Poisson, Irwin–Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub‐Gaussian and sub‐exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.