Over the years, numerous rank estimators for factor models have been proposed in the literature. This article focuses on information criterion-based rank estimators and investigates their consistency in rank selection. The gap conditions serve as necessary and sufficient conditions for rank estimators to achieve selection consistency under the general assumptions of random matrix theory. We establish a unified theorem on selection consistency, presenting the gap conditions for information criterion-based rank estimators with a unified formulation. To validate the theorem's assertion that rank selection consistency is solely determined by the gap conditions, we conduct extensive numerical simulations across various settings. Additionally, we undertake supplementary simulations to explore the strengths and limitations of information criterion-based estimators by comparing them with other types of rank estimators.

翻译：多年来，文献中已提出众多因子模型的秩估计量。本文聚焦于基于信息准则的秩估计量，并研究其秩选择的一致性。在随机矩阵理论的一般假设下，间隙条件是秩估计量实现选择一致性的充分必要条件。我们建立了一个关于选择一致性的统一定理，以统一形式给出了基于信息准则的秩估计量的间隙条件。为验证该定理关于秩选择一致性仅由间隙条件决定的论断，我们在多种设定下进行了广泛的数值模拟。此外，我们还通过与其他类型秩估计量的比较，开展了补充模拟以探究基于信息准则的估计量的优势与局限。