Multivariate linear regressions are widely used statistical tools in many applications to model the associations between multiple related responses and a set of predictors. To infer such associations, it is often of interest to test the structure of the regression coefficients matrix, and the likelihood ratio test (LRT) is one of the most popular approaches in practice. Despite its popularity, it is known that the classical $\chi^2$ approximations for LRTs often fail in high-dimensional settings, where the dimensions of responses and predictors $(m,p)$ are allowed to grow with the sample size $n$. Though various corrected LRTs and other test statistics have been proposed in the literature, the fundamental question of when the classic LRT starts to fail is less studied, an answer to which would provide insights for practitioners, especially when analyzing data with $m/n$ and $p/n$ small but not negligible. Moreover, the power performance of the LRT in high-dimensional data analysis remains underexplored. To address these issues, the first part of this work gives the asymptotic boundary where the classical LRT fails and develops the corrected limiting distribution of the LRT for a general asymptotic regime. The second part of this work further studies the test power of the LRT in the high-dimensional setting. The result not only advances the current understanding of asymptotic behavior of the LRT under alternative hypothesis, but also motivates the development of a power-enhanced LRT. The third part of this work considers the setting with $p>n$, where the LRT is not well-defined. We propose a two-step testing procedure by first performing dimension reduction and then applying the proposed LRT. Theoretical properties are developed to ensure the validity of the proposed method. Numerical studies are also presented to demonstrate its good performance.

中文翻译：

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多元线性回归中的似然比检验：从低维到高维

多元线性回归是许多应用中广泛使用的统计工具，用于对多个相关响应和一组预测变量之间的关联进行建模。为了推断这种关联，测试回归系数矩阵的结构通常很有趣，而似然比检验 (LRT) 是实践中最流行的方法之一。尽管它很受欢迎，但众所周知，LRT 的经典 $\chi^2$ 近似通常在高维设置中失败，其中允许响应和预测变量 $(m,p)$ 的维度随着样本大小 $ 增长n$。尽管文献中已经提出了各种修正的 LRT 和其他测试统计数据，但对经典 LRT 何时开始失败的基本问题研究较少，这个答案将为从业者提供见解，特别是在分析 $m/n$ 和 $p/n$ 小但不可忽略的数据时。此外，轻轨在高维数据分析中的功率性能仍未得到充分探索。为了解决这些问题，这项工作的第一部分给出了经典 LRT 失败的渐近边界，并为一般渐近机制开发了 LRT 的修正极限分布。本工作的第二部分进一步研究了轻轨在高维设置下的测试能力。该结果不仅促进了当前对替代假设下 LRT 渐近行为的理解，而且还推动了功率增强型 LRT 的发展。这项工作的第三部分考虑了 $p>n$ 的设置，其中 LRT 没有明确定义。我们提出了一个两步测试程序，首先执行降维，然后应用建议的 LRT。理论性质的发展，以确保所提出的方法的有效性。还提出了数值研究以证明其良好的性能。