This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization (and improvement) of the celebrated Chernoff method. At its heart, it is based on deriving a new class of composite nonnegative martingales, with strong connections to betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform competing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, with and without replacement.
翻译:本文针对从受约束的观测中估算一个未知值的经典问题,得出了信任间隔(CI)和时间一致性信任序列(CS),用于估算从受约束的观测中得出一个未知值的典型问题。我们提出了一个得出浓度界限的一般方法,可以被看作是著名的Chernoff方法的概括化(和改进 ) 。 在文件的心脏上,它是基于产生一个新的非阴性混合马丁鱼的类别,与赌注和混合物方法有着密切的联系。我们展示了如何将这些想法扩大到抽样而不替换,另一个经过大量研究的问题。 在所有情况下,我们的界限都适应了未知的差异,在经验上大大超越了基于Hoffding或经验性Bernstein不平等及其最近超martingal一般化的相互竞争方法。简言之,我们为四个基本问题建立了新的状态,即CS和CI作为约束手段,有和无替代手段。