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**Model checking for parametric single-index models: a dimension reduction model-adaptive approach.**
*(English)*
Zbl 1414.62131

Summary: Local smoothing testing based on multivariate non-parametric regression estimation is one of the main model checking methodologies in the literature. However, the relevant tests suffer from the typical curse of dimensionality, resulting in slow rates of convergence to their limits under the null hypothesis and less deviation from the null hypothesis under alternative hypotheses. This problem prevents tests from maintaining the level of significance well and makes tests less sensitive to alternative hypotheses. In the paper, a model adaptation concept in lack-of-fit testing is introduced and a dimension reduction model-adaptive test procedure is proposed for parametric single-index models. The test behaves like a local smoothing test, as if the model were univariate. It is consistent against any global alternative hypothesis and can detect local alternative hypotheses distinct from the null hypothesis at a fast rate that existing local smoothing tests can achieve only when the model is univariate. Simulations are conducted to examine the performance of our methodology. An analysis of real data is shown for illustration. The method can be readily extended to global smoothing methodology and other testing problems.