Testing model assumptions in functional regression models

In the functional regression model where the responses are curves new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L^2-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples bootstrap versions of the tests improve the quality of the approximation of the nominal level