论文标题
关于威尔克现象的阶段过渡
On the Phase Transition of Wilk's Phenomenon
论文作者
论文摘要
Wilk的定理为可能性比测试提供通用的卡方近似值，被广泛用于许多科学假设测试问题中。对于维度不断增加的现代数据集，研究人员发现，传统的Wilk对可能性比率测试统计量的现象通常会失败。尽管在高维度设置中提出了新的近似值，但仍然缺乏有关如何在常规和新提出的近似值之间进行选择的明确统计准则，尤其是对于中度维数据。为了解决这个问题，我们在对多元平均值和协方差结构的流行测试下为Wilk现象开发了必要和充分的相变条件。此外，我们通过推导其渐近偏见来深入分析卡方近似的准确性。这些结果可能会提供有关在科学实践中使用卡方近似值的有用见解。
Wilk's 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 Wilk's 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 Wilk's 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.
