Testing Mutually Exclusive Hypotheses for Multi-Response Regressions
提出一种自适应模型检验方法，用于检验参数多响应回归中响应向量至多一个坐标与预测向量相关的原假设，通过分解为互斥子假设并构建混合检验，适用于高维数据。
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.