Asymptotic inference for high-dimensional data
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高维数据的渐近推断
- 研究问题
- 当数据维度相对样本量很大、参数数量可随机增长且可能存在缺失时，均值估计的一致性、渐近正态性及均值向量检验如何建立？
- 方法与思路
- 作者构建无限维框架，在不同分量尾部条件下研究均值估计，并由渐近结果构造单样本和双样本检验，辅以模拟、方法比较和实际数据分析。
- 材料陈述的贡献
- 摘要称给出均值估计联合一致性的精确条件，澄清并改进部分已有结果，建立渐近正态性及检验极限分布，并形成有渐近依据的分析方法。
- 阅读关注点
- 应核对随机维度的增长限制、缺失数据机制、尾部条件与所用函数空间，以及模拟中偏离理论条件时检验表现如何变化。
- 阅读边界
- 仅依据版本未注明的 OpenAlex 索引摘要；缺失机制、具体条件和比较数值未提供，模拟中的稳健表现不能替代一般有效性证明。
原文摘要
In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is, allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide precise conditions for the joint consistency of the estimators of the mean. In the process, we clarify and improve some of the recent consistency results that appeared in the literature. An important aspect of the work presented is the development of asymptotic normality results for these models. As a consequence, we construct different test statistics for one-sample and two-sample problems concerning the mean vector and obtain their asymptotic distributions as a corollary of the infinite-dimensional results. Finally, we use these theoretical results to develop an asymptotically justifiable methodology for data analyses. Simulation results presented here describe situations where the methodology can be successfully applied. They also evaluate its robustness under a variety of conditions, some of which are substantially different from the technical conditions. Comparisons to other methods used in the literature are provided. Analyses of real-life data is also included.