AI 中文总结
本文针对各领域广泛应用但统计特性未被充分研究的块结构隐变量模型，推导其可识别性条件，提出拉格朗日型优化公式，建立估计量的误差界与渐近分布，经模拟和实证分析验证了理论结果。
AI 中文摘要
本文研究块结构隐变量模型，其中观测变量根据其与潜在隐变量的关系被划分为不同的块。这类块结构在心理学、教育学、经济学和遗传学等多个领域中普遍存在。尽管应用广泛，但这些模型的基础统计特性仍在很大程度上未被探索。在本研究中，我们对块结构隐变量模型进行了全面的统计分析。特别地，我们首先推导了不同块设计下的模型可识别性条件。此外，我们研究了这些可识别性约束下的最大似然估计。为了适应与各种块配置相关的复杂约束，我们为该约束非凸优化问题引入了一种拉格朗日型公式，并证明其最优解与原问题的最优解一致。该公式是理解不同块结构下约束估计量行为的关键工具。在此基础上，我们建立了约束最大似然估计量的精确非渐近误差界和渐近分布。我们还提出了一种获取该估计量的计算框架，并为算法输出建立了理论特性。我们的理论发现通过模拟研究和实证数据分析得到了验证。
英文摘要
This paper studies block structured latent variable models, in which observed variables are grouped into distinct blocks based on their relationships with the underlying latent variables. These block structures are prevalent in various fields such as psychology, education, economics, and genetics. Despite their widespread applications, the fundamental statistical properties of these models remain largely unexplored. In this work, we present a comprehensive statistical analysis of the block structured latent variable models. In particular, we first derive conditions for model identifiability across various block designs. Furthermore, we investigate the maximum likelihood estimation under these identifiability constraints. To accommodate these intricate constraints associated with various block configurations, we introduce a Lagrangian-type formulation for the constrained nonconvex optimization problem and show that its optimum coincides with that of the original problem. This formulation serves as a critical tool for understanding the behavior of the constrained estimator under various block structures. Building on that, we establish sharp non-asymptotic error bounds and asymptotic distributions of the constrained maximum likelihood estimator. We also propose a computational framework to obtain the estimator and establish theoretical properties for the algorithm output. Our theoretical findings are validated through simulation studies and empirical data analyses.
