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.
翻译：信任序列是一个信任间隔序列, 在一个没有限制的时间范围内统一有效 。 我们的工作发展了宽度为零的信任序列, 在非参数条件下, 以非无防疫覆盖保障 。 我们在指数浓度的Cram\'er- Chernoff 方法、 迭代对数法( LIL) 和顺序概率比率测试 -- 我们的信任序列是第一个时间格式的扩展; 对第二个空间提供紧凑、 不便利的描述; 将第三个环境普遍化为非参数环境, 包括亚加西和伯恩斯坦条件、 自我正常化进程 和矩阵马丁格莱斯之间 。 我们通过得出以 LIL 率增长的经验- Bernstein 方法, 以及随机矩阵总和的最大树本价值的新颖的LIL LIL 方法, 展示了我们的证据技术的普遍性。 最后, 我们运用了我们的方法对内曼- 鲁宾 潜在结果模型下的样本平均处理效果进行调和估计。
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