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.