We construct a goodness-of-fit test for the Functional Linear Model with Scalar Response (FLMSR) with responses Missing At Random (MAR). For that, we extend an existing testing procedure for the case where all responses have been observed to the case where the responses are MAR. The testing procedure gives rise to a statistic based on a marked empirical process indexed by the randomly projected functional covariate. The test statistic depends on a suitable estimator of the functional slope of the FLMSR when the sample has MAR responses, so several estimators are proposed and compared. With any of them, the test statistic is relatively easy to compute and its distribution under the null hypothesis is simple to calibrate based on a wild bootstrap procedure. The behavior of the resulting testing procedure as a function of the estimators of the functional slope of the FLMSR is illustrated by means of several Monte Carlo experiments. Additionally, the testing procedure is applied to a real data set to check whether the linear hypothesis holds.

翻译：针对响应随机缺失（MAR）情况下的标量响应泛函线性模型（FLMSR），我们构建了拟合优度检验。为此，我们将现有针对完全观测响应情形的检验程序扩展至响应MAR情形。该检验程序基于由随机投影泛函协变量索引的带符号经验过程构建统计量。由于检验统计量依赖于MAR响应样本中FLMSR泛函斜率的合适估计量，我们提出并比较了若干估计方案。采用任一估计方案时，检验统计量的计算均相对简便，且基于野自助法可便捷地校准其零假设分布。通过多项蒙特卡洛实验，我们阐明了检验程序性能作为FLMSR泛函斜率估计量的函数关系。此外，将该检验程序应用于实际数据集以验证线性假设是否成立。