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) + ε, 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 ε. 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 χ2approximation 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.