Goodness-of-fit tests for high-dimensional parametric multiresponse regressions

This study investigates goodness-of-fit testing in the context of high-dimensional parametric multiresponse regression, where the numbers of responses and parameters may diverge as the sample size increases. Particular attention is given to ordinary differential equation models, whose solutions can be interpreted as examples of multiresponse regression. To address the challenges posed by these models, we introduce two novel tests: a new global smoothing test and a new local smoothing test. The former exhibits a normal weak limit under the null hypothesis, representing a significant improvement over classical counterparts with intractable limiting null distributions in fixed-dimensional scenarios. The latter exhibits a dimension-agnostic property under certain conditions. Moreover, an increase in the number of responses can enhance the sensitivity of both tests to alternative models. Under specific regularity conditions, these tests can detect local alternatives distinct from the null hypothesis at rates faster than the fastest possible rates achievable by classical tests in fixed-dimensional cases. Numerical studies are conducted to evaluate the performance of the proposed tests and demonstrate their efficacy in real data analysis.