A classical problem of statistical inference is the valid specification of a model that can account for the statistical dependencies between observations when the true structure is dense, intractable, or unknown. To address this problem, a new variance identity is presented, which is closely related to the Moulton factor. This identity does not require the specification of an entire covariance structure and instead relies on the choice of two summary constants. Using this result, a weak law of large numbers is also established for additive statistics and common variance estimators under very general conditions of statistical dependence. Furthermore, this paper proves a sharper version of Hoeffding's inequality for symmetric and bounded random variables under these same conditions of statistical dependence. Put otherwise, it is shown that, under relatively mild conditions, finite sample inference is possible in common settings such as linear regression, and even when every outcome variable is statistically dependent with all others. All results are extended to estimating equations. Simulation experiments and an application to climate data are also provided.

翻译：统计推断的一个经典问题是在真实结构密集、难解或未知时，如何有效指定能够解释观测值间统计依赖关系的模型。为解决该问题，本文提出了一个与莫尔顿因子密切相关的新方差恒等式。该恒等式无需指定完整的协方差结构，仅依赖于两个汇总常数的选择。基于这一结果，本文还在统计依赖的非常一般条件下，建立了加性统计量和常见方差估计量的弱大数定律。此外，本文在相同统计依赖条件下，证明了针对对称有界随机变量的更精确的霍夫丁不等式。换言之，在相对温和的条件下，即使每个结果变量与其他所有变量均存在统计依赖时，线性回归等常见场景仍可实现有限样本推断。所有结果均被推广至估计方程。本文还提供了模拟实验及气候数据应用案例。