We consider the problem of testing for the nullity of conditional expectations of Hilbert space-valued random variables. We allow for conditioning variables taking values in finite or infinite Hilbert spaces. This testing problem occurs, for instance, when checking the goodness-of-fit or the effect of some infinite-dimensional covariates in regression models for functional data. Testing the independence, between a finite dimensional variable and a functional one, is another example that could be treated in our framework. We propose a new test based on kernel smoothing. The test statistic is asymptotically standard normal under the null hypothesis provided the smoothing parameter tends to zero at a suitable rate. The one-sided test is consistent against any fixed alternative, as well as against local alternatives a la Pitman and uniformly against classes of regular alternatives approaching the null hypothesis. In particular, we show that neither the dimension of the outcome nor the dimension of the functional covariates influences the theoretical power of the test against such alternatives. Simulation experiments and a real data application using a variable-domain functional regression model illustrate the performance of the new test.

中文翻译：

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检验功能协变量的显着性

我们考虑检验希尔伯特空间值随机变量的条件期望是否为零的问题。我们允许在有限或无限希尔伯特空间中取值的条件变量。例如，在检查函数数据的回归模型中的拟合优度或某些无限维协变量的影响时，就会出现这种测试问题。测试有限维变量和函数变量之间的独立性是我们框架中可以处理的另一个例子。我们提出了一种基于核平滑的新测试。如果平滑参数以合适的速率趋于零，则检验统计量在零假设下是渐近标准正态的。单方面测试与任何固定的替代方案一致，以及反对本地替代品 a la Pitman 并统一反对接近零假设的常规替代品类别。特别是，我们表明结果的维度和功能协变量的维度都不会影响针对此类替代方案的测试的理论能力。模拟实验和使用可变域函数回归模型的真实数据应用说明了新测试的性能。