AI 中文总结
研究希尔伯特空间中不完全数据的统计检验，提出建模不完全数据的方法，识别其独立同分布结构，借助数学结果分析统计程序，还给出检验不同假设的通用新颖概念及正态性拟合优度检验示例。
AI 中文摘要
我们考虑基于可分希尔伯特空间中不完全观测值的统计检验，其维度可能很大甚至无穷。一般的希尔伯特空间设置允许现代应用中出现的各种数据类型，特别是高维和函数型数据。可能的希尔伯特空间检验问题包括拟合优度、对称性、同质性和独立性。我们提出一种用于对不完全数据建模的方法，涵盖实际中的几个问题，如带有缺失项的超高维随机向量或部分观测的随机过程。我们在不完全数据中识别出一种特定结构（独立同分布），借助合适的数学结果（如大数定律和中心极限定理）来分析统计程序。此外，针对这种情况提出了一个用于检验不同假设的通用且新颖的概念，并以正态性拟合优度检验为例进行了概述。
英文摘要
We consider statistical testing on the basis of incomplete observations with values in a separable Hilbert space, where the dimension is possibly large or even infinite. The general Hilbert space setting allows various data types as they arise in modern applications, in particular high dimensional and functional data. Possible Hilbert space testing problems are goodness-of-fit, symmetry, homogeneity and independence. We present an approach for modeling incomplete data that covers several problems in practice, e.g., ultra high dimensional random vectors with missing entries or partially observed stochastic processes. We identify a specific structure (independent and identically distributed) in the incomplete data that enables the analysis of statistical procedures with the help of suitable mathematical results (e.g., laws of large numbers and central limit theorems). Additionally, a general and novel concept for testing different hypotheses in this situation is suggested and sketched for the example of testing goodness-of-fit for normality.
