  We propose a procedure for testing the linearity of a scalar-on-function
regression relationship. To do so, we use the functional generalized additive
model (FGAM), a recently developed extension of the functional linear model.
For a functional covariate X(t), the FGAM models the mean response as the
integral with respect to t of F{X(t),t} where F is an unknown bivariate
function. The FGAM can be viewed as the natural functional extension of
generalized additive models. We show how the functional linear model can be
represented as a simple mixed model nested within the FGAM. Using this
representation, we then consider restricted likelihood ratio tests for zero
variance components in mixed models to test the null hypothesis that the
functional linear model holds. The methods are general and can also be applied
to testing for interactions in a multivariate additive model or for testing for
no effect in the functional linear model. The performance of the proposed tests
is assessed on simulated data and in an application to measuring diesel truck
emissions, where strong evidence of nonlinearities in the relationship between
the functional predictor and the response are found.
