A goodness-of-fit test for the Functional Linear Model with Scalar Response (FLMSR) with responses Missing at Random (MAR) is proposed in this paper. The test statistic relies on a marked empirical process indexed by the projected functional covariate and its distribution under the null hypothesis is calibrated using a wild bootstrap procedure. The computation and performance of the test rely on having an accurate estimator of the functional slope of the FLMSR when the sample has MAR responses. Three estimation methods based on the Functional Principal Components (FPCs) of the covariate are considered. First, the simplified method estimates the functional slope by simply discarding observations with missing responses. Second, the imputed method estimates the functional slope by imputing the missing responses using the simplified estimator. Third, the inverse probability weighted method incorporates the missing response generation mechanism when imputing. Furthermore, both cross-validation and LASSO regression are used to select the FPCs used by each estimator. Several Monte Carlo experiments are conducted to analyze the behavior of the testing procedure in combination with the functional slope estimators. Results indicate that estimators performing missing-response imputation achieve the highest power. The testing procedure is applied to check for linear dependence between the average number of sunny days per year and the mean curve of daily temperatures at weather stations in Spain.

翻译：本文提出了一种针对响应随机缺失的标量响应函数线性模型的拟合优度检验。检验统计量依赖于一个由投影函数协变量索引的标记经验过程，其原假设下的分布通过野马自举过程进行校准。当样本存在响应随机缺失时，检验的计算与性能依赖于FLMSR函数斜率的精确估计。本文考虑了基于协变量函数主成分的三种估计方法：第一，简化方法直接丢弃缺失响应观测值来估计函数斜率；第二，插值方法利用简化估计量对缺失响应进行插补来估计函数斜率；第三，逆概率加权方法在插补时纳入缺失响应生成机制。此外，使用交叉验证和LASSO回归来选择每种估计量所采用的FPCs。通过蒙特卡洛实验分析了检验程序与函数斜率估计量结合时的表现。结果表明，执行缺失响应插补的估计量具有最高的检验功效。该检验程序被应用于检验西班牙气象站年均晴天天数与日均气温均值曲线之间的线性依赖关系。