A conventional regression model for functional data involves expressing a response variable in terms of the predictor function. Two assumptions, that (i) the predictor function and the error are independent and (ii) the relationship between the response variable and the predictor function takes functional linear model, are usually added to the model. Checking the validation of these two assumptions is fundamental to statistic inference and practical applications. We develop a test procedure to check these assumptions simultaneously based on generalized distance covariance. We establish the asymptotic theory for the proposed test under null and alternative hypotheses, and provide a bootstrap procedure to obtain the critical value of the test. The proposed test is consistent against all alternatives provided that the semimetrics related to the generalized distance are strong negative, and can be readily generalized to other functional regression models. We explore the finite sample performance of the proposed test by using both simulations and real data examples. The results illustrate that the proposed method has favorable performance compared with the competing method.

用于功能数据的常规回归模型涉及根据预测函数表达响应变量。通常在模型中添加两​​个假设，即（i）预测函数和误差是独立的，以及（ii）响应变量和预测函数之间的关系采用函数线性模型。检查这两个假设的有效性是统计推断和实际应用的基础。我们开发了一种测试程序，可以基于广义距离协方差同时检查这些假设。我们在零假设和替代假设下为拟议的测试建立了渐近理论，并提供了一个自举程序来获得测试的临界值。如果与广义距离有关的半度量是强负值，并且可以很容易地推广到其他功能回归模型，则所提出的测试与所有替代方法都一致。我们通过使用仿真和实际数据示例来探索所提出测试的有限样本性能。结果表明，与竞争方法相比，该方法具有良好的性能。