LIKELIHOOD RATIO TEST IN MULTIVARIATE LINEAR REGRESSION: FROM LOW TO HIGH DIMENSION
LIKELIHOOD RATIO TEST IN MULTIVARIATE LINEAR REGRESSION: FROM LOW TO HIGH DIMENSION
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DOI：
10.5705/ss.202019.0056
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发表时间：
2021-07-01
影响因子：
1.4
通讯作者：
Xu, Gongjun
中科院分区：
文献类型：
--
作者：
He, Yinqiu;Jiang, Tiefeng;Xu, Gongjun
多元线性回归被广泛用于对多个相关反应和一组预测因子之间的关联进行建模。为了推断这种关联，研究人员经常测试回归系数矩阵的结构，通常使用似然比检验（LRT）。尽管它们很受欢迎，但已知LRT的经典chi(2)近似在高维设置中失败，在高维设置中，响应的维度和预测因子（m, p）允许随着样本量n的增长而增长。尽管已经提出了各种校正的LRT和其他测试统计量，但很少有研究检查经典LRT何时开始失败的重要问题。这个问题的答案将为从业者提供见解，特别是在分析m/n和p/n很小但不可忽略的数据时。此外，LRT在高维数据分析中的作用还有待进一步研究。为了解决这些问题，本工作的第一部分确定了经典LRT失效的渐近边界，并开发了具有一般渐近区域的LRT的修正极限分布。本研究的第二部分考察了轻轨在高维环境中的能力。除了推进当前对替代假设下LRT渐近行为的理解外，这些结果还激励了更强大的LRT的发展。本工作的第三部分考虑了p > n的设置，其中轻轨没有很好地定义。我们建议采用两步测试程序。首先，我们执行降维，然后应用建议的LRT。理论推导证明了该方法的有效性，仿真结果表明了该方法的有效性。
Multivariate linear regressions are widely used to model the associations between multiple related responses and a set of predictors. To infer such associations, researchers often test the structure of the regression coefficients matrix, usually using a likelihood ratio test (LRT). Despite their popularity, classical chi(2) approximations for LRTs are known to fail in high-dimensional settings, where the dimensions of the responses and the predictors (m, p) are allowed to grow with the sample size n. Although various corrected LRTs and other test statistics have been proposed, few studies have examined the important question of when the classic LRT starts to fail. An answer to this would provide insights for practitioners, especially when analyzing data in which m/n and p/n are small, but not negligible. Moreover, the power of the LRT in high-dimensional data analyses remains under-researched. To address these issues, the first part of this work determines the asymptotic boundary at which the classical LRT fails, and develops a corrected limiting distribution for the LRT with a general asymptotic regime. The second part of this work examines the power of the LRT in high-dimensional settings. In addition to advancing the current understanding of the asymptotic behavior of the LRT under an alternative hypothesis, these results motivate the development of a more powerful LRT. The third part of this work considers the setting in which p > n, where the LRT is not well defined. We propose a two-step testing procedure. First, we perform a dimension reduction, and then we apply the proposed LRT. Theoretical properties are developed to ensure the validity of the proposed method, and simulations demonstrate that the method performs well.