Testing Mutually Exclusive Hypotheses for Multi-Response Regressions

This article proposes an adaptive-to-model test to check the null hypothesis with no more than one coordinate of the response vector relating to the predictor vector in parametric multi-response regressions. To this end, we decompose the null hypothesis into several mutually exclusive sub-null hypotheses and suggest a model identification to construct an adaptive-to-sub-null hypothesis test tackling their mutual exclusiveness, and an adaptive-to-regression test handling the regression function mis-specification. The final test combines a further model identification to be an adaptive-to-model hybrid of these two tests. It has the Chi-square weak limit under the null hypothesis even when the dimensions of the response and the predictor vectors increase with the sample size and is omnibus. We conduct a systematic analysis of the significance level maintenance and power performance of the test to reveal its different sensitivity rates of convergence to different sub-local alternatives distinct from the null hypothesis. This is a significant distinction against any existing model checking problems for regressions. Further, the proposed model identifications can also assist in identifying the responses with nonconstant regressions and testing their mis-specification. Numerical studies include simulations to examine the finite sample performances and to illustrate real data analyses for two datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本文提出一种自适应模型检验方法，用于在参数化多响应回归中检验原假设——响应向量中与预测向量相关的坐标不超过一个。为此，我们将原假设分解为若干互斥的子原假设，并提出模型识别方法：一方面构建自适应子原假设检验以应对子原假设的互斥性，另一方面构建自适应回归检验以处理回归函数的误设问题。最终检验通过进一步的模型识别，将这两种检验融合为自适应模型混合检验。即使响应向量和预测向量的维度随样本量增大，该检验在原假设下仍具有卡方弱极限，且是万能的。我们对该检验的显著性水平保持能力和检验功效表现进行系统分析，揭示其对与原假设不同的各子局部备择假设具有不同的收敛敏感率。这与现有回归模型检验问题相比具有显著差异。此外，所提出的模型识别方法还可辅助识别具有非恒定回归的响应变量，并检验其回归函数的误设。数值研究包括模拟实验以验证有限样本表现，以及对两个数据集的真实数据分析示例。本文的补充材料可在线获取，包括用于复现研究的材料的标准化描述。