  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.
