论文标题
对具有潜在变量的模型的可能性比测试的注释
A Note on Likelihood Ratio Tests for Models with Latent Variables
论文作者
论文摘要
似然比测试（LRT）广泛用于比较嵌套潜在变量模型的相对拟合。按照Wilks的定理，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LRT，在比较模型下，具有自由度的$χ^2 $ - 分布等于两个嵌套模型之间的自由参数数量的差异。然而，对于具有潜在变量的模型，例如因子分析，结构方程模型和随机效应模型，经常发现$χ^2 $近似不存在。在本说明中，我们展示了Wilks定理的规律性条件如何使用带有潜在变量的模型的三个示例违反。此外，还为LRT提供了更一般的理论，该理论为这些LRT提供了正确的渐近理论。该一般理论最初是在Chernoff（1954）建立的，并在Van der Vaart（2000）和Drton（2009）中进行了讨论，但似乎没有得到足够的关注。我们用三个例子说明了这一一般理论。
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 $χ^2$-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 $χ^2$ 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 (1954) and discussed in both van der Vaart (2000) and Drton (2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.
