Variance estimation is important for statistical inference. It becomes non-trivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these nuisance structures. This paper develops a general framework for the estimation of the long-run variance for time series with non-constant means. The building blocks are difference statistics. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first asymptotically optimal estimator is derived. Our proposed estimator is theoretically proven to be invariant to arbitrary mean structures, which may include trends and a possibly divergent number of discontinuities.

翻译：差异估算对于统计推论很重要。 当观测被串联依赖结构和时间变化的平均值结构掩盖时,它就变得非三边性。 现有的方法要么忽略,要么次巧妙地处理这些麻烦结构。 本文用非连续手段为时间序列的长期差异估算制定一个总框架。 构件是差异统计。 拟议的估计数据类别非常笼统,足以涵盖现有的许多估计数据。 调查了一致性的必要和充分条件。 提出了第一个非同步的最佳估计数据。 我们提议的估计数据在理论上证明对任意平均结构是不可变的, 这可能包括趋势以及可能存在差异的不连续性。