Abstract A lack-of-fit test for functional regression models is proposed. The test is based on the fact that checking the no-effect of a functional covariate is equivalent to checking the nullity of the conditional expectation of the error term given a sufficiently rich set of projections of that covariate. The idea then is to search the projection that is, in some sense, the least favorable for the null hypothesis. Finally, it remains to perform a nonparametric check of the nullity of the conditional expectation of the residuals of the regression given the selected least favorable projection. For the search of a least favorable projection and the nonparametric check we use a kernel-based approach. As a result, the test statistic is a quadratic form based on univariate kernel smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect general departures from the model. The error term of the regression could present heteroscedasticity of unknown form. The law of the functional covariate need not be known. The test could be implemented quite easily and performs well in simulations and real data applications.

中文翻译：

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针对一般替代方案测试功能回归模型中的失配

摘要 提出了一种函数回归模型的失拟检验。该测试基于这样一个事实，即检查函数协变量的无影响等效于检查给定该协变量的一组足够丰富的投影的误差项的条件期望的无效性。然后的想法是搜索在某种意义上对零假设最不利的投影。最后，给定选定的最不利投影，仍然要对回归残差的条件期望的无效性进行非参数检查。为了搜索最不利的投影和非参数检查，我们使用基于内核的方法。因此，检验统计量是基于单变量核平滑的二次形式，渐近临界值由标准正态定律给出。该测试能够检测与模型的一般偏差。回归的误差项可能呈现未知形式的异方差性。不需要知道函数协变量的定律。该测试可以很容易地实现，并且在模拟和实际数据应用中表现良好。