Testing mutually exclusive hypotheses for multi-response regressions

This paper 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 non-constant regressions and testing their mis-specification. Numerical studies include simulations to examine the finite sample performances and to illustrate real data analyses for two data sets.

本文提出一种模型自适应检验（adaptive-to-model test），用于检验参数化多响应回归（parametric multi-response regressions）中“响应向量至多存在一个分量与预测向量相关”的原假设。为此，我们将该原假设拆解为若干个互斥子原假设，并提出一种模型识别方法，分别构建用于处理子原假设间互斥性的子原假设自适应检验（adaptive-to-sub-null hypothesis test），以及用于应对回归函数设定误差的回归自适应检验（adaptive-to-regression test）。最终检验结合了进一步的模型识别步骤，形成了这两种检验的模型自适应混合检验。即便响应向量与预测向量的维度随样本量增长，该检验在原假设下仍服从卡方弱极限（chi-square weak limit），且属于整体检验。我们对该检验的显著性水平维持性能与检验效能开展了系统性分析，揭示了其针对各类与原假设相异的子局部备择假设的收敛灵敏度差异。这一特性与现有所有回归模型检验问题均存在显著区别。此外，本文提出的模型识别方法还可用于识别存在非常数回归的响应分量，并对其设定误差进行检验。数值实验包含模拟实验，用于检验有限样本表现，并通过两个数据集展示实际数据分析案例。