Time-uniform, nonparametric, nonasymptotic confidence sequences
Time-uniform, nonparametric, nonasymptotic confidence sequences
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DOI：
10.1214/20-aos1991
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发表时间：
2018-10
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影响因子：
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通讯作者：
Steven R. Howard;Aaditya Ramdas;Jon D. McAuliffe;J. Sekhon
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文献类型：
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作者：
Steven R. Howard;Aaditya Ramdas;Jon D. McAuliffe;J. Sekhon
置信序列是在无限时间范围内一致有效的置信区间序列。我们的工作发展的置信度序列的宽度为零，与非渐近覆盖的非参数条件下的保证。我们绘制指数浓度的Cram\'er-Schiff方法，迭代对数法（LIL）和序贯概率比检验之间的连接-我们的置信序列是第一个时间一致的扩展;提供第二个紧密的，非渐近的特征;并将第三个推广到非参数设置，包括次高斯和伯恩斯坦条件，自归一化过程和矩阵鞅。我们说明了我们的证明技术的一般性，通过推导出一个numerical-Bernstein界增长的LIL率，以及一个新的上LIL的随机矩阵的总和的最大特征值。最后，我们将我们的方法应用于协方差矩阵估计和估计样本平均治疗效果下的奈曼-鲁宾潜在的结果模型。
A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cram\'er-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL), and the sequential probability ratio test---our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes, and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.