Testing for no effect in the spatial functional linear regression model

We consider a functional linear regression model with a real -valued response and a functional random variable with its derivative as covariates. We are interested in testing the null hypothesis of no covariates effect using a spatially dependent sample. We propose two test statistics which take into account the proximity between sites and we establish the asymptotic normality of cross covariance operator between both interest variables. From this result, we derive asymptotic distributions of these both statistics. Then, we illustrate our test procedure by means of a simulation study.