经验标准化和的尾界

摘要：

和的指数尾界在统计学中起着重要的作用，但$t$ -统计量的例子表明，当需要从数据中估计总体参数时，指数尾衰减可能会丢失。然而，事实证明，如果学生化伴随着以适当的方式估计位置参数，则$t$统计量恢复指数尾部行为。受这个例子的启发，本文分析了经验标准化求和的其他方法，并为以下设置建立了亚高斯或更接近正常的尾部边界：正态观测的标准化与学生化对比，指数族观测的标准化与对数似然比统计量，以及对于来自具有未知对称中心的对称分布的观测值，通过自归一化进行标准化。后者的标准化产生了一种新的扫描统计异方差数据的渐近功率分析的情况下，观察有一个对数凹分布。

英文摘要：

Exponential tail bounds for sums play an important role in statistics, but the example of the $t$-statistic shows that the exponential tail decay may be lost when population parameters need to be estimated from the data. However, it turns out that if Studentizing is accompanied by estimating the location parameter in a suitable way, then the $t$-statistic regains the exponential tail behavior. Motivated by this example, the paper analyzes other ways of empirically standardizing sums and establishes tail bounds that are sub-Gaussian or even closer to normal for the following settings: Standardization with Studentized contrasts for normal observations, standardization with the log likelihood ratio statistic for observations from an exponential family, and standardization via self-normalization for observations from a symmetric distribution with unknown center of symmetry. The latter standardization gives rise to a novel scan statistic for heteroscedastic data whose asymptotic power is analyzed in the case where the observations have a log-concave distribution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。