Testing Hypotheses in the Functional Linear Model

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of R and the response is scalar. The response is modelled as Y =Ψ( X )+ ɛ , where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in R . The random input X is independent from the noise ɛ . In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X . We introduce two test statistics based on the norm of the empirical cross-covariance operator of ( X , Y ). The first test statistic relies on a χ ² approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X . The test procedures can be applied to check a given relationship between X and Y . The method is illustrated through a simulation study.