Wilks’ theorem, which offers universal chi-squared approximations for likelihood ratio tests, is widely used in many scientific hypothesis testing problems. For modern datasets with increasing dimension, researchers have found that the conventional Wilks’ phenomenon of the likelihood ratio test statistic often fails. Although new approximations have been proposed in high-dimensional settings, there still lacks a clear statistical guideline regarding how to choose between the conventional and newly proposed approximations, especially for moderate-dimensional data. To address this issue, we develop the necessary and sufficient phase transition conditions for Wilks’ phenomenon under popular tests on multivariate mean and covariance structures. Moreover, we provide an in-depth analysis of the accuracy of chi-squared approximations by deriving their asymptotic biases. These results may provide helpful insights into the use of chi-squared approximations in scientific practices.

中文翻译：

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威尔克斯现象的相变

Wilks 定理为似然比检验提供通用卡方近似值，广泛用于许多科学假设检验问题。对于维度不断增加的现代数据集，研究人员发现，似然比检验统计量的常规 Wilks 现象经常失败。尽管已经在高维设置中提出了新的近似值，但仍然缺乏关于如何在传统和新提出的近似值之间进行选择的明确统计指南，尤其是对于中维数据。为了解决这个问题，我们在多变量均值和协方差结构的流行测试下为威尔克斯现象开发了必要和充分的相变条件。而且，我们通过推导卡方近似的渐近偏差，对卡方近似的准确性进行了深入分析。这些结果可能为在科学实践中使用卡方近似提供有用的见解。