完备性是否必要？非识别线性模型中的估计
Is completeness necessary? Estimation in nonidentified linear models
AI总结：
本文探讨了非识别线性模型中正则化估计器的完备性需求，提出了一种在识别失败情况下仍能提供非标准渐近分布的理论框架。
摘要：
现代数据分析越来越依赖于通过灵活的高维或非参数机器学习方法来估计模型，其中结构参数的识别往往具有挑战性且无法检验。在线性设置中，这种识别依赖于完备性条件，该条件要求高维矩阵或算子的非奇异性和可能在有限样本或甚至总体层面失败。正则化估计器提供了解决方案，通过使结构或平均结构函数的一致估计成为可能，有时即使在识别失败的情况下也是如此。我们证明在这些情况下，渐近分布可能是非标准的。我们开发了正则化估计器的全面理论，其中包括高维岭正则化、梯度下降和主成分分析（PCA）等方法。结果通过高维和非参数工具变量回归进行了示例，并通过模拟实验得到了支持。
英文摘要：
Modern data analysis depends increasingly on estimating models via flexible high-dimensional or nonparametric machine learning methods, where the identification of structural parameters is often challenging and untestable. In linear settings, this identification hinges on the completeness condition, which requires the nonsingularity of a high-dimensional matrix or operator and may fail for finite samples or even at the population level. Regularized estimators provide a solution by enabling consistent estimation of structural or average structural functions, sometimes even under identification failure. We show that the asymptotic distribution in these cases can be nonstandard. We develop a comprehensive theory of regularized estimators, which include methods such as high-dimensional ridge regularization, gradient descent, and principal component analysis (PCA). The results are illustrated for high-dimensional and nonparametric instrumental variable regressions and are supported through simulation experiments.
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