A test for heteroscedasticity in functional linear models

We propose a new test to validate the assumption of homoscedasticity in a functional linear model. We consider a minimum distance measure of heteroscedasticity in functional data, which is zero in the case where the variance is constant and positive otherwise. We derive an explicit form of the measure, propose an estimator for the quantity, and show that an appropriately standardized version of the estimator is asymptotically normally distributed under both the null (homoscedasticity) and alternative hypotheses. We extend this result for residuals from functional linear models and develop a bootstrap diagnostic test for the presence of heteroscedasticity under the postulated model. Moreover, our approach also allows testing for “relevant” deviations from the homoscedastic variance structure and constructing confidence intervals for the proposed measure. We investigate the performance of our method using extensive numerical simulations and a data example.