This paper makes 3 contributions. First, it generalizes the Lindeberg\textendash Feller and Lyapunov Central Limit Theorems to Hilbert Spaces by way of $L^2$. Second, it generalizes these results to spaces in which sample failure and missingness can occur. Finally, it shows that satisfaction of the Lindeberg\textendash Feller Condition in such spaces guarantees the consistency of all inferences from the partial functional data with respect to the completely observed data. These latter two results are especially important given the increasing attention to statistical inference with partially observed functional data. This paper goes beyond previous research by providing simple boundedness conditions which guarantee that \textit{all} inferences, as opposed to some proper subset of them, will be consistently estimated. This is shown primarily by aggregating conditional expectations with respect to the space of missingness patterns. This paper appears to be the first to apply this technique.
翻译:本文提出了3项意见。 首先,本文将Lindeberg\ textendash Feller和Lyapunov中央限制理论概括为$L $2$,将Libert空间的Lyapunov Central Limit Theorems 概括为Hilbert Spaces。 其次,本文将这些结果概括为样本失败和缺失可能发生的空间。 最后,该文件表明,Lindeberg\ textendash Felleral Condition在这些空间的满意度保证了与完全观察到的数据有关的部分功能数据的所有推断的一致性。 后两项结果特别重要, 因为人们越来越关注部分观测到的功能数据的统计推论。 本文超越了先前的研究范围, 提供了简单的约束条件, 保证对“ textititit{all} 推论, 而不是其中某些适当的分类, 将会得到一致的估计。 这主要表现在将关于缺失空间的有条件期望综合。 该文件似乎是应用这一技术的第一个。