The likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks' theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a [Formula: see text] distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison. For models with latent variables such as factor analysis, structural equation models and random effects models, however, it is often found that the [Formula: see text] approximation does not hold. In this note, we show how the regularity conditions of Wilks' theorem may be violated using three examples of models with latent variables. In addition, a more general theory for LRT is given that provides the correct asymptotic theory for these LRTs. This general theory was first established in Chernoff (J R Stat Soc Ser B (Methodol) 45:404-413, 1954) and discussed in both van der Vaart (Asymptotic statistics, Cambridge, Cambridge University Press, 2000) and Drton (Ann Stat 37:979-1012, 2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.

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关于具有潜在变量的模型的似然比检验的说明

似然比检验 (LRT) 广泛用于比较嵌套潜在变量模型的相对拟合。根据威尔克斯定理，LRT 是通过将 LRT 统计量与其在受限模型下的渐近分布进行比较来进行的，[公式：见正文] 分布的自由度等于两个嵌套模型之间的自由参数数量的差异比较之下。然而，对于因子分析、结构方程模型和随机效应模型等具有潜在变量的模型，经常发现[公式：见正文]近似不成立。在本笔记中，我们展示了如何使用具有潜在变量的模型的三个示例违反 Wilks 定理的规律性条件。此外，给出了更一般的 LRT 理论，为这些 LRT 提供了正确的渐近理论。这个一般理论首先在 Chernoff (JR Stat Soc Ser B (Methodol) 45:404-413, 1954) 中建立，并在 van der Vaart (Asymptotic statistics, Cambridge, Cambridge University Press, 2000) 和 Drton (Ann Stat 37 :979-1012, 2009)，但似乎并没有受到足够的重视。我们用三个例子来说明这个一般理论。