Testing the goodness of fit of a functional autoregressive model

The proposed Goodness-of-Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space H. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space H, with subordinator given by the marginal probability of the involved Autoregressive Hilbertian process (ARH(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The simulation study provided in the Appendix illustrates the finite-sample performance of the testing procedure under different families of alternatives.