Abstract The functional linear quantile regression model is widely used to characterize the relationship between a scalar response and a functional covariate. Most existing research results are based on a correct assumption that the response is related to the functional predictor through a linear model for given quantile levels. This paper focuses on investigating the adequacy check of the functional linear quantile regression model. We propose a nonparametric U-process test statistic based on the functional principal component analysis. It is proved that the test statistic follows a normal distribution asymptotically under the null hypothesis and diverges to infinity for any misspecified models. Therefore, the test is consistent against any fixed alternative. Moreover, it is shown that the test has asymptotic power one for the local alternative hypothetical models converging to the null hypothesis at the rates n − 1 2 . The finite sample properties of the test statistic are illustrated through extensive simulation studies. A real data set of 24 hourly measurements of ozone levels in Sacramento, California is analyzed by the proposed test.

中文翻译：

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检查函数线性分位数回归模型的充分性

摘要 泛函线性分位数回归模型被广泛用于表征标量响应与泛函协变量之间的关系。大多数现有研究结果都基于正确的假设，即响应通过给定分位数水平的线性模型与函数预测器相关。本文重点研究函数线性分位数回归模型的充分性检验。我们提出了基于功能主成分分析的非参数 U 过程检验统计量。事实证明，检验统计量在零假设下渐近服从正态分布，并且对于任何错误指定的模型都会发散到无穷大。因此，该测试与任何固定替代方案一致。而且，结果表明，对于以 n − 1 2 的速率收敛到零假设的局部替代假设模型，该检验具有渐近幂 1。检验统计量的有限样本属性通过广泛的模拟研究来说明。拟议的测试分析了加利福尼亚州萨克拉门托 24 小时臭氧水平测量值的真实数据集。